Build your students understanding of the difference between dividing a whole number by a unit fraction and multiplying a whole number by a unit fraction with this set of task cards and printables. The double set of task cards presents problem situations and visual models and provide opportunities for analysis of the models, written reflection, and practice with varied ways to represent multiplication and division situations. With this “print-and-go” resource, you’ll have everything you need to develop, strengthen, and assess your students’ understanding of representing multiplication and division of whole numbers by unit fractions.
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Common Core State Standards for Mathematics addressed:
Numbers and Operations – Fractions (5.NF)
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
• Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. (5.NF.6)
• Solve real world problems involving division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. (5.NF.7c)
• 2 reference sheets
• two sets of 32 task cards
• task card answer sheets and keys
• 4 assessment activities
• rubric and answer key for assessment activities
I designed this set of resources to help my students develop an understanding of the difference between dividing by a fraction and multiplying by a fraction. The difference between multiplication situations and division situations can seem straightforward when only whole numbers are involved, but multiplying and dividing with fractions really throws kids for a loop. Much of multiplying and dividing with fractions seem counterintuitive. For example, since finding 1/3 of 12 involves splitting 12 into thirds, this naturally seems like it would be 12 ÷ 1/3 when it is actually 12 x 1/3. Similarly, dividing 12 by 1/3 results in a number much larger than 12, it seems more like multiplying than dividing. In fact, when we first began working with dividing whole numbers by unit fractions, some of my students believed that 12 ÷ 1/3 and 12 x 1/3 would have the same answer!
The focus of these materials is not simply finding the product or quotient of a whole number and unit fraction but understanding what it means
to multiply and divide by a fraction.
About the Cards
This is a double set of task cards, labeled as Set B1 and Set B2. All of the cards present a problem situation that relates to either multiplying a whole number by a unit fraction or dividing a whole number by a unit fraction. Set B1 asks students to describe the relationship between the model and the problem, while Set B2 presents a problem situation and a corresponding model and asks students to solve the problem. The problems and models on Set B1 and Set B2 are, by and large, identical. Since the two sets have different purposes – B1 focuses on communication of mathematical thinking while B2 focuses on actually solving the word problems –, you can use both sets, despite their having the same problems.
The first sixteen cards in Set B1 present students with a situation and a model, and they are asked to describe how each model represents the given situation. The second half of the set also feature situations and models, but not every model matches the given situation. For these cards, students have to decide if the model actually matches the situation, and then explain why or why not. Because of the nature of the cards, the answer sheets provided are different than those included with my other task card sets. The first answer sheet has space for a student’s name and the date, and sets of lines for students to respond to four cards. The next answer sheet also features lines for four cards, and you can copy as many of these as you need, depending on the number of cards your students will use. If you prefer, you can save copies by having your students simply use notebook paper (or their journals) to write their answers to each card. The scoring guide for Set B1 includes a suggested rubric, using a 3 point scale, as well as a sample answer for each of the 32 cards.
Set B2 presents the same problems found on Set B1 as well as a corresponding model, and the students are simply asked to solve the problem. There are two different answer sheets provided for use with Set B2. One answer sheet contains blanks for students to record the answer to the problem. The other answer sheet presents students with two expressions, one using multiplication and one using division (e.g., 8 ÷ ¼ and 8 x ¼) and the students have to select the expression that matches the story. Choose the answer sheet that best matches your students need, or use one answer sheet at one point in the year and the other at a later point in the year when you need to review.
Using the Cards
Since the two sets have different purposes – B1 focuses on communication of mathematical thinking while B2 focuses on actually solving the word problems –, you can use both sets, despite their having the same problems. You might have your students work through the cards in Set B1 when they are first exploring with the concept of dividing and multiplying with unit fractions, and then work with the cards in Set B2 later on in the year as a review. You might mix the sets together, having your students alternate between writing about the models and solving the problems.
There are lots of ways in which you can implement the task cards. 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 all the task cards in one session. 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. This last method is how I used Set B1 in my own classroom. Because the students were required to do writing, the cards in this set took a little longer than the typical task cards. I would give out a sheet, we would review one or two of the task cards together to talk through the relationship between the story and the model and writing a group answer, and then my students would complete the other cards on the sheet on their own.
Reinforcing the Concept
The printables consist of two graphic reference sheet and four different one-page worksheets. The first reference sheet presents two problem situations and corresponding models, as well as some questions directed at the students aimed at pushing them to reason about the difference between the problems and the relationship between the models and the problems. I designed this resource not as a standard reference sheet with rules and procedures, but as an interactive tool, providing a springboard for a class discussion and some journal writing about comparative situations. Have your students glue the reference in their journal, project it using your computer or document camera, then work through the examples and questions. Your students can use the think-pair-share strategy to discuss their thinking prompted by the questions, and then respond in their math notebooks or math journals. This scaffolded approach to communication – student-to-student, followed by student-to-class, followed by student writing – is a great way to build students’ thinking about mathematical concepts, to involve all students in the communication process, and to provide a rich bank of ideas so that when it comes time to write, all students will have something substantive to write about.
The second reference sheet presents two problems and corresponding model, as well as an explanation of why division matches one of the problems and multiplication matches the other. Students are given two other problems and asked to consider which of the problems can be represented by division and which by multiplication.
Assessing Student Understanding
The four provided assessment activities can be used to evaluate student understanding of using to models when multiplying and dividing whole numbers by unit fractions. The first two pair of assessment activities present story problems and models and the students have to identify which model relates to the given problem. The second pair of assessments presents three story problems and asks students to create their own models to represent the problems. The activities in each pairs are formatted similarly, and have similar types of questions, though the numbers on each are different. You can use these activity pages in a variety of ways. You could give one or two as a pre-test (perhaps combining assessment activities a and c), then teach your lesson and allow students to practice with the task cards, and then give the others as an independent post-test. You could also have the students work on the task cards, then complete one of the worksheets as guided practice with yourself, a partner, or a small group, and then give the second worksheet as an independent assessment. The worksheets could also be given as homework, center assignments, or any other purpose that fits your teaching style or classroom routines
For more practice with fraction computation concepts, please check out the other related resources I have available –
Fraction Puzzlers - fraction story problems task cards + printables (set b)
Foxy Fractions - adding/subtracting unlike denominators task cards + printables
Name That Equation - fraction multiplication task cards + printables set
Scaling Fractions bundle - ppts, task cards, game, and printables
I hope your students enjoy these resources and are able to build their proficiency with fraction computation. – Dennis McDonald