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Build your students’ understanding of mixed numbers and improper fractions with this set of resources that focuses on *reasoning*, not rules and procedures. The 4 reference sheets and 32 task cards will help your students grow more flexible in their understanding of fractional representations, seeing that 3 & 1/2 isn’t just equal to 7/2, but also 2 & 3/2 and 1 & 5/2. Extend their practice, or assess their level of understanding, with the 4 assessment tasks. With this “print-and-go” resource, you’ll have everything you need to develop, strengthen, and assess your students’ understanding of mixed numbers and improper fractions.

Included:

• 4 reference sheets

• 32 task cards

• task card answer sheet and key

• 8 self-checking “answer cards”

• 4 assessment activities

• rubric and answer key for assessment activities

I designed this set of resources to help develop my fourth grade students’ flexibility with improper fractions and mixed numbers. When I am teaching improper fractions and mixed numbers, I do not teach the students a series of steps, e.g., multiply the denominator by the whole number, add the numerator, and make this number the new numerator. I have found that when students are simply taught such a rule, later on many of them invariably remember some of the steps, but not all of them, and often they apply them in the wrong order. I end up with some kids multiplying the numerator by the denominator and adding the whole number, while others multiply the numerator by the whole number and then add the denominator. Some of the students who do learn the procedure have been able to rename in both directions, but when I have asked them questions that requires more than simply multiplying and adding, such as, “Between what two whole numbers does 27/4 fall?” or “What is the value of a in the expression 3 & 2/5 = 2 & a/5?”, I often have been greeted with blank stares.

**About the Cards**

The tasks on these cards require more than simply using the procedure to rename an improper fraction as a mixed number and vice versa. They won’t be asked to simply rename 4 & 2/3 into 14/3, but recognize that 4 & 2/3 is equal to 3 & 5/3 and 2 & 8/3 and 1 & 11/3 as well as 14/3. The cards have mostly multiple-choice questions, with a few open-ended short answer questions mixed in. To add to the challenge level, the multiple-choice questions don’t all have just one answer. Some have only one answer while others have two or more correct answers. Your students will really have to think through each question to be sure that they identified every possible correct answer, and as they do, they will be actively composing and decomposing, naming and renaming fractions. The problems on the cards provide opportunities for building fractions with manipulatives and/or representing them with visual models, such as number lines, as well as equations. By the time your students have finished working with these resources, they will have developed a fluency and flexibility with mixed numbers and improper fractions that will help them as they move into computing with mixed numbers.

*Please check out the preview to see all of the materials up close!*

**Using the Cards**

The denominators on these cards are limited to the ones identified by the Common Core Standards for Grade 4 as “limiters”: 2, 3, 4, 5, 6, 8, 10, and 12. One advantage of using only these denominators is that these fractional units are commonly used on commercially-made fraction bars, fraction circles, and fraction squares. This allows for easy differentiation within the math class. If you have students that are still building an understanding of fractional sizes and relationships, you may choose to give them a set of fraction manipulatives with which to complete the cards. Your students who already have a strong understanding of fractional sizes can complete the cards without the concrete representation.

As an alternative to manipulatives, you can have your students use the included number line reference sheets. They present halves, thirds, fourths, and fifths through six wholes, and sixths, eighths, tenths, and twelfths through four wholes. Have your students glue the number line reference sheets in their journals and then have your students discuss the patterns they see in the improper fractions on the number lines. Since not all of the lines are labeled, you can have them identify the improper fractions that belong on the unlabeled lines. See if the students can use the number lines to identify equivalent improper fractions and mixed numbers. There is so much to do with these number lines, and the more your students examine and discuss them, the more they will internalize the fractional relationships represented on the number lines - and the more flexible and fluid they will be when working with mixed numbers and improper fractions.

There are lots of ways in which you can implement the task cards beyond the suggestions above. You can have the students work on them independently, working through the task cards on their own. The students can work on them in pairs or small groups, completing a given number of cards in one or more sessions. You can use them in centers, having the students complete 6-8 task cards a day over the course of the week. You can even use them as a variation of “problem of the day”, giving each student 1 sheet of 4 cards to glue in their journals and solve, one sheet per day for eight days.

**Reinforcing the Concept**

You may choose to introduce or follow-up the cards with the included reference sheets. Besides the two sheets of number lines described earlier, there is also a reference sheet that defines vocabulary related to improper fractions and mixed numbers and demonstrates the use of a number line to rename improper fractions into mixed numbers, and vice versa. The fourth reference sheet also uses number lines, in this case to illustrate the various ways to represent an amount greater than 1 whole (e.g., 4 & 2/3 = 3 & 5/3 = 2 & 8/3…). Most of the task cards require students to be able to regroup one or more wholes in mixed numbers, and so this reference sheet can be the jumping off point for a discussion and exploration with varied fractional representations. Both of these reference sheets feature open-ended questions that can be used as prompts for group discussions and/or journal writing. Your students can use the journal inserts as guides while they work on the cards, as well as when they complete other tasks that relate to mixed numbers and improper fractions.

**Assessing Student Understanding**

The four provided activity sheets can be used to evaluate student understanding of mixed number/improper fraction relationships. The first two assessment activities features questions similar to those on the cards. The second pair of assessment activities focus on analysis and reasoning. They each present the work and explanation of a hypothetical student who made one or more errors when renaming an improper fraction or mixed number. For each activity, your students will have to examine the work and then explain what is incorrect about the student’s reasoning. The activities in each pair are formatted similarly, and have similar types of questions, though the numbers on each are different. The similar formatting allows for easy use as pre/post assessments, though you can use these activities in a variety of ways – guided practice, homework, center assignments, or any other purpose that fits your teaching style or classroom routines.

For more practice with the fraction concepts, please check out the other related resources I have available –

**Panda Power - fraction/whole number equivalence resource bundle**

Decomposing Fractions - activity card & printables bundle

Monkey Mania & Jumping Giraffes – equivalent fraction games + task cards bundle

I hope your students enjoy these resources and are able to build their proficiency with fractions. – Dennis McDonald

Included:

• 4 reference sheets

• 32 task cards

• task card answer sheet and key

• 8 self-checking “answer cards”

• 4 assessment activities

• rubric and answer key for assessment activities

I designed this set of resources to help develop my fourth grade students’ flexibility with improper fractions and mixed numbers. When I am teaching improper fractions and mixed numbers, I do not teach the students a series of steps, e.g., multiply the denominator by the whole number, add the numerator, and make this number the new numerator. I have found that when students are simply taught such a rule, later on many of them invariably remember some of the steps, but not all of them, and often they apply them in the wrong order. I end up with some kids multiplying the numerator by the denominator and adding the whole number, while others multiply the numerator by the whole number and then add the denominator. Some of the students who do learn the procedure have been able to rename in both directions, but when I have asked them questions that requires more than simply multiplying and adding, such as, “Between what two whole numbers does 27/4 fall?” or “What is the value of a in the expression 3 & 2/5 = 2 & a/5?”, I often have been greeted with blank stares.

The tasks on these cards require more than simply using the procedure to rename an improper fraction as a mixed number and vice versa. They won’t be asked to simply rename 4 & 2/3 into 14/3, but recognize that 4 & 2/3 is equal to 3 & 5/3 and 2 & 8/3 and 1 & 11/3 as well as 14/3. The cards have mostly multiple-choice questions, with a few open-ended short answer questions mixed in. To add to the challenge level, the multiple-choice questions don’t all have just one answer. Some have only one answer while others have two or more correct answers. Your students will really have to think through each question to be sure that they identified every possible correct answer, and as they do, they will be actively composing and decomposing, naming and renaming fractions. The problems on the cards provide opportunities for building fractions with manipulatives and/or representing them with visual models, such as number lines, as well as equations. By the time your students have finished working with these resources, they will have developed a fluency and flexibility with mixed numbers and improper fractions that will help them as they move into computing with mixed numbers.

The denominators on these cards are limited to the ones identified by the Common Core Standards for Grade 4 as “limiters”: 2, 3, 4, 5, 6, 8, 10, and 12. One advantage of using only these denominators is that these fractional units are commonly used on commercially-made fraction bars, fraction circles, and fraction squares. This allows for easy differentiation within the math class. If you have students that are still building an understanding of fractional sizes and relationships, you may choose to give them a set of fraction manipulatives with which to complete the cards. Your students who already have a strong understanding of fractional sizes can complete the cards without the concrete representation.

As an alternative to manipulatives, you can have your students use the included number line reference sheets. They present halves, thirds, fourths, and fifths through six wholes, and sixths, eighths, tenths, and twelfths through four wholes. Have your students glue the number line reference sheets in their journals and then have your students discuss the patterns they see in the improper fractions on the number lines. Since not all of the lines are labeled, you can have them identify the improper fractions that belong on the unlabeled lines. See if the students can use the number lines to identify equivalent improper fractions and mixed numbers. There is so much to do with these number lines, and the more your students examine and discuss them, the more they will internalize the fractional relationships represented on the number lines - and the more flexible and fluid they will be when working with mixed numbers and improper fractions.

There are lots of ways in which you can implement the task cards beyond the suggestions above. You can have the students work on them independently, working through the task cards on their own. The students can work on them in pairs or small groups, completing a given number of cards in one or more sessions. You can use them in centers, having the students complete 6-8 task cards a day over the course of the week. You can even use them as a variation of “problem of the day”, giving each student 1 sheet of 4 cards to glue in their journals and solve, one sheet per day for eight days.

You may choose to introduce or follow-up the cards with the included reference sheets. Besides the two sheets of number lines described earlier, there is also a reference sheet that defines vocabulary related to improper fractions and mixed numbers and demonstrates the use of a number line to rename improper fractions into mixed numbers, and vice versa. The fourth reference sheet also uses number lines, in this case to illustrate the various ways to represent an amount greater than 1 whole (e.g., 4 & 2/3 = 3 & 5/3 = 2 & 8/3…). Most of the task cards require students to be able to regroup one or more wholes in mixed numbers, and so this reference sheet can be the jumping off point for a discussion and exploration with varied fractional representations. Both of these reference sheets feature open-ended questions that can be used as prompts for group discussions and/or journal writing. Your students can use the journal inserts as guides while they work on the cards, as well as when they complete other tasks that relate to mixed numbers and improper fractions.

The four provided activity sheets can be used to evaluate student understanding of mixed number/improper fraction relationships. The first two assessment activities features questions similar to those on the cards. The second pair of assessment activities focus on analysis and reasoning. They each present the work and explanation of a hypothetical student who made one or more errors when renaming an improper fraction or mixed number. For each activity, your students will have to examine the work and then explain what is incorrect about the student’s reasoning. The activities in each pair are formatted similarly, and have similar types of questions, though the numbers on each are different. The similar formatting allows for easy use as pre/post assessments, though you can use these activities in a variety of ways – guided practice, homework, center assignments, or any other purpose that fits your teaching style or classroom routines.

For more practice with the fraction concepts, please check out the other related resources I have available –

Decomposing Fractions - activity card & printables bundle

Monkey Mania & Jumping Giraffes – equivalent fraction games + task cards bundle

I hope your students enjoy these resources and are able to build their proficiency with fractions. – Dennis McDonald

Total Pages

23 pages

Answer Key

Included with rubric

Teaching Duration

N/A

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