Multiplying makes things bigger, right? This is what my kids always think – as I’m sure yours do as well. Of course, this causes quite a problem when they start to work with multiplying fractions, where the “rules” they have learned for multiplication don’t seem to apply!
This bundle – including an instructional ppt, 2 sets of 32 task cards, reference sheets, and assessment activities, and an interactive ppt quiz – is a comprehensive resource designed to introduce, build, and extend to the concept of multiplication as scaling, and it provides everything you need in one “print-and-go” package. After you have used these materials with your kids, they will have a much better understanding of how multiplication can both increase and decrease the value of a number.
This bundle consists of materials sold separately –
Predicting Products - fraction multiplication as scaling task cards + printables
Growing & Shrinking – scaling fractions ppt, task cards, and printables
Gain Some, Lose Some – fraction multiplication as scaling game and printables
Save $$$ and receive an exclusive interactive ppt quiz, only available here, by purchasing this bundle!
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.
Interpret multiplication as scaling (resizing), by:
• Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. (5.NF.5a)
• Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by
a fraction less than 1 results in a product less than the given number; and relating the principle of fraction equivalence a/b = (n x a)/(n x b) to the effect of multiplying a/b by 1. (5.NF.5b)
Fraction concepts are a major focus of the Common Core State Standards for Math in intermediate grades, and the expectations of elementary students in terms of what they understand about fractions is (in many instances) significantly more advanced than what was expected of them pre-Common Core. By the end of fifth grade, students are expected to have mastered multiplication and division of fractions, concepts that, before now, many students were not even exposed to until middle school.
One of the challenges I have found with helping students master fraction concepts is that some rules and procedures for analyzing and working with fractions are the same as those used with whole numbers, and others are different. Bigger numbers mean bigger value, right? Not when they are the denominators! My students faced another example of the counter-intuitive nature of fractions when we began working with fraction multiplication. For years, they have been told that when you multiply two numbers, you get a bigger number. [Of course, this is not technically true even of all whole numbers, but that’s a math misconception for another day!] Welcome to multiplication with fractions and mixed numbers, where sometimes the product is greater than both factors, sometimes the product is greater than just one factor, and sometimes the product is actually less than both factors. Throw in the fact that you can multiply by a fraction and have a number equal to one of the factors, and you have a recipe for some confused students.
• 20-slide powerpoint presentation
• 5 different reference sheets
• 2 sets of 32 task cards – 64 cards in all
• 2 sets of 8 self-checking “answer cards”
• task card answer sheets and keys
• game & recording sheet
• 7 assessment activities and scoring guide/rubric
• 15-slide interactive powerpoint quiz
This product is a ZIP file containing 2 PPTX files and 3 PDF files. For directions about how to “unzip” the files, TpT provides instructions here
Growing and Shrinking
All of the materials in this set use the metaphor of a “sizing potion”, some of which increase the size of an object (x 3, x 6) and some of which decrease the size of an object (x 1/2 , x 3/4). The goal was for my students to understand that when you multiply a given amount by a number greater than 1, the resulting product is larger (because it has “grown”), and when you multiply the same amount by a number less than 1, the resulting product is smaller (because it has “shrunk”).
Begin with the powerpoint, which uses graphics and animations to bring to life the concept of multiplication as scaling. The animations illustrate objects (in this case, dinosaurs) growing when given a potion that uses a whole number and shrinks with given a potion that uses a fraction less than one. All of the animations are timed to appear automatically, with arrows appearing on each slide when the slide needs to be advanced. A number of the slides present discussion questions that will allow your class to have a small-group and/or whole-group conversation about the concepts presented.
Follow-up the powerpoint by providing your students with one or both of the included reference sheets and the 32 task cards. The task cards use the same metaphor of “sizing potions”, continuing where the powerpoint leaves off. Each card presents the students with a given potion and a description of the size of a hypothetical pet (for example, a snake that is ¾ of a foot long). The students have to figure out from the label on the potion whether it is a growing potion or a shrinking potion and identify whether the animal’s new size will be greater than its original size or less than its original size.
The provided assessment activities can be used to evaluate student understanding of multiplication as scaling. Each of the two-page activities uses a mix of multiple-choice, open-ended, and written response questions. The two activities are formatted similarly, and have similar types of questions, though the numbers on each are different. I designed them this way so they could be easily used as a pre/post assessment. However, you can use these activity pages in a variety of ways – guided practice, paired work, homework, center assignments, or any other purpose that fits your teaching style or classroom routines.
NOTE: For these materials, as well as the others in this set, the students do not
need to be able to actually multiply the numbers to determine the size of the product. In fact, the standard itself requires students to be able to identify and explain how a product compares to the size of factors without calculating the numbers. When I taught this standard and used these materials, I had not yet taught students how to multiply fractions. I think that was actually helpful because lacking a knowledge of the procedure, the students couldn’t just multiply the numbers and compare – they had to actually use reasoning.
This set of task cards, reference sheets, and assessment activities extend the concepts introduced in the Growing and Shrinking
set, with the focus on evaluating expressions, equations, and inequalities that use fractions and whole numbers as factors.
I designed these materials to help my students practice using the size of factors to identify the size of the product of those factors. Each card presents the students with a multiplication equation with an unknown product. The equations use proper fractions, improper fractions, and whole numbers as factors. The types of factors used in the equations are limited to: a) proper fraction x proper fraction; b) whole number x proper fraction; c) proper fraction x improper fraction equal to one: and, d) proper fraction x improper fraction greater than one. Students are asked to decide whether the product is less than both factors, the product is less than one factor and greater than the other factor, or the product is equal to one of the factors.
Included among the printables are a full-page graphic reference sheet and a foldable reference sheet. These references are different from the one that supports the “Growing and Shrinking” ppt, and they make an excellent introduction to or review of the concepts addressed by the cards. The two provided assessment activities can be used to evaluate student understanding of the size relationship between factors & products in multiplication equations that use fractions as factors. Both activities are a mix of multiple-choice questions, open-ended questions that have more than one answer, and an opportunity for students to explain their thinking in writing. These activities are formatted similarly, and have similar types of questions, though the numbers on each are different. I designed them this way so they could be easily used as a pre/post assessment. However, you can use these activity pages in a variety of ways – guided practice, paired work, homework, center assignments, or any other purpose that fits your teaching style or classroom routines.
Gain Some, Lose Some
Once your students have had a chance to build an understanding of how the size of factors affect the size of the product, use this game to provide quick, repeated, and fun practice with the concept. It requires few materials – chips or tokens and a paper clip or plastic spinner –, and the directions for the game are printed on the board itself, so all you need is the one sheet. My kids really enjoyed playing the game – especially because of the element of being able to have an opponent lose chips – and the variety of numbers on the spinners (whole numbers, improper fractions and proper fractions) forced my students to consider all of the “rules” that govern multiplication with fractions. With four different spinners and eight numbers on each spinner, your kids can play this again and again without repeating combinations of factors.
To play, you will need to provide each player with 10 chips or tokens as well as a cup of more tokens that will be the “pot.” A player spins twice to get two factors. Then, they gain or lose chips based on whether the product of the two factors would be less than both factors, greater than both factors, greater than one factor and less than the other, or equal to one factor. The way that players gain and lose chips matches the relationship between the factors and products. If the factors are both less
than one, for instance, the player will lose
two chips, reinforcing the idea that the product is less
than both factors. If the factors are both greater
than one, however, the player will gain
two chips, reinforcing the idea that the product is greater
than both factors.
Included with the game is a recording sheet that you can have your students fill out as they play to show the factors they spun and their “prediction” about the size of the product.
Follow up the game with any of the three different assessment tasks to evaluate their understanding of product/factor relationships when multiplying with fractions. Each of the tasks requires students to evaluate multiplication expressions, equations, and/or inequalities based on how the size of their products compare to the size of their factors. For each activity, the students are asked to explain their thinking in writing, providing an opportunity to evaluate your students’ ability to communicate clearly and effectively using math vocabulary. Of course, these tasks needn’t be used simply as assessments. You can use them as guided review, paired practice, homework assignments – any way that suits your students’ needs! Keys and a rubric are provided for each of the tasks, and each assessment has a total of 10 points, making for easy conversion to a percentage grade.
Please check out the preview for a close-up look at some of the materials that are part of this bundle. I think you will find them to be useful resources in building your students’ understanding of multiplication as scaling.– Dennis McDonald