GRADE SUPPLEMENT Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Includes H H H H H H H Activity 1: Multiplying Fractions on the Geoboard Activity 2: Quick Sketches & Fraction Arrays Activity 3: Fraction Word Problems & Arrays Independent Worksheet 1: Pizza Party Planning Independent Worksheet 2: Brownie Dessert Independent Worksheet 3: Multiplying Fractions with Arrays Independent Worksheet 4: Company’s Coming! A10.1 A10.9 A10.13 A10.17 A10.19 A10.21 A10.23 Skills & Concepts H H H H H H H H Add fractions with like denominators Write an equation showing a fraction a ⁄b as the sum of a number of the unit fraction ⁄b Write an equation to show a fraction as the sum of other fractions with the same denominator Demonstrate an understanding that a fraction a ⁄b is a multiple of the unit fraction ⁄b Write an equation showing that a fraction a ⁄b is the product of a × ⁄b Multiply a fraction by a whole number Solve word problems in which fractions are multiplied by a whole number Use a visual area model to multiply a whole number by a fraction P201309 Bridges in Mathematics Grade Supplement Set A10 Number & Operations: Multiplying Whole Numbers by Fractions The Math Learning Center, PO Box 12929, Salem, Oregon 97309 Tel 800 575–8130 © 2013 by The Math Learning Center All rights reserved Prepared for publication on Macintosh Desktop Publishing system Printed in the United States of America P201309 The Math Learning Center grants permission to classroom teachers to reproduce blackline masters in appropriate quantities for their classroom use Bridges in Mathematics is a standards-based K–5 curriculum that provides a unique blend of concept development and skills practice in the context of problem solving It incorporates the Number Corner, a collection of daily skill-building activities for students The Math Learning Center is a nonproit organization serving the education community Our mission is to inspire and enable individuals to discover and develop their mathematical conidence and ability We offer innovative and standards-based professional development, curriculum, materials, and resources to support learning and teaching To ind out more, visit us at www.mathlearningcenter.org Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Set A10 H Activity ACTIVITY Multiplying Fractions on the Geoboard Overview You’ll need Students re-examine problems from Bridges Unit 6: Session 3, and consider how the relationship between areas on a geoboard might be represented using multiplication Students then consider additional models on the geoboard in which fractions are multiplied by a whole number H Fractions on a Geoboard: Multiplying Fractions, pages A10.5–A10.7 (run a class set.) Skills & Concepts H Paper/pen for use on the display H Geoboards and geobands or Geoboard App (online or iPad) for display H Geoboards and geobands for students, optional a H Write an equation showing a fraction ⁄b as the sum of a number of the unit fraction ⁄b (CCSS 4.NF.3) H Colored pencils, optional H Express a fraction as the sum of other fractions with the same denominator in more than one way (CCSS 4.NF.3b) Note When you represent the symbolic form for a fraction, please use a horizontal bar H Demonstrate an understanding that a fraction a ⁄b is a multiple of the unit fraction ⁄b (CCSS 4.NF.4a) H Write an equation showing that a fraction a ⁄b is the product of a × ⁄b (CCSS 4.NF.4a) H Use an area model to multiply a fraction by a whole number (CCSS 4.NF.4b) H Explore fraction relationships and operations using a visual model H Find equivalent fractions Instructions for Multiplying Fractions on the Geoboard Distribute copies of Fractions on a Geoboard: Multiplying Fractions and display the top portion Give students time to build all the regions on their geoboards Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Blackline Run a class set NAME DATE Fractions on a Geoboard Multiplying Fractions In the illustration below, the whole geoboard has an area of A B C D E Write C + C = B on the whiteboard Have student pairs discuss if the equation is true Ask for a volunteer to share Then ask student pairs to consider using multiplication to show the relationship between regions B and C © The Math Learning Center Bridges in Mathematics Grade Supplement • A10.1 Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Activity Multiplying Fractions on the Geoboard cont After students have had a chance to share observations with a partner, ask for volunteers to share with the class Teacher Jesse, where you see multiplication in the relationship between C and B? Jesse Well, if two Cs equal B, that’s like saying that × C = B Lauren And since C is 1⁄8, you could write × 1⁄8 = 1⁄4 Max I see it as 1⁄4 = × 1⁄8 Complete information in the first row as students write on their own sheets Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Blackline Run a class set NAME DATE Fractions on a Geoboard Multiplying Fractions In the illustration below, the whole geoboard has an area of A B C D E Look at the visual models below and think about what you notice about the relationships between the regions on the geoboard How could you express your observations using multiplication? Explain your thinking in words and record your observations in a multiplication equation Visual Model Explanation Multiplication Equation a There are C’s in Part B 2x = B C C b Look at the next visual model With student input, review ways that the relationship between D and A could be represented with addition Compile a list of equivalent equations on the board 16 + 161 + 161 + 161 + 161 + 161 + 161 + 16 = + 16 = + 162 + 162 + 16 = 16 16 16 Teacher Does anyone know another way we could write ⁄16? Mia One half You can see it on the geoboard is half of 16 A10.2 • Bridges in Mathematics Grade Supplement © The Math Learning Center Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Activity Multiplying Fractions on the Geoboard cont Ask students to help you write an explanation such as “There are D’s in section A.” By writing the explanation, students will begin to more explicitly see fraction equations as a sum of the number of a unit fraction Then ask each student to record one way that relationship between D and A could be expressed using multiplication Share with partners Then ask for volunteers to share the multiplication equation Marin If 1⁄16 + 1⁄16 + 1⁄16 + 1⁄16 + 1⁄16 + 1⁄16 + 1⁄16 + 1⁄16 = 8⁄16, it’d be a lot easier to write × 1⁄16 = 8⁄16 Josh Mine was like Marin’s But I said that × 1⁄16 = 1⁄2 is half of 16, like Mia said Marin ⁄16 and 1⁄2 are the same thing Cybil How can × 1⁄16 be 1⁄2? Teacher Josh, could you show us on the geoboard? Josh Sure Look at the D’s that are shaded in That makes sixteenths But the entire part that’s shaded in shows 1⁄2 of the whole geoboard So sixteenths is 1⁄2 Or × 1⁄16 = 1⁄2 Continue having students look at parts in relationship to the entire area of the geoboard Ask them to write explanations and multiplication equations to represent the shaded portion of the geoboard Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Blackline Run a class set Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Blackline Run a class set NAME NAME DATE DATE Fractions on a Geoboard Multiplying Fractions cont Fractions on a Geoboard Multiplying Fractions Visual Model In the illustration below, the whole geoboard has an area of E A E E E C B C D E Look at the visual models below and think about what you notice about the relationships between the regions on the geoboard How could you express your observations using multiplication? Explain your thinking in words and record your observations in a multiplication equation Visual Model Explanation Explanation Multiplication Equation d The whole geoboard can be split into 32 E’s C can be split into E’s 4x 32 = The whole geoboard can be split into B’s Two shaded parts is 42 or 21 2x = 2x = 6x = or = 16 or e B B Multiplication Equation a There are C’s in Part B 2x = f B C C C C C C C C The whole geoboard can be split into C’s of the parts are shaded b D D D D D D D D There are D’s in Part A A 8x 16 = 16 8x 16 = 4x = g D D D D The whole geoboard can be split into 16 D’s of 16 are shaded 4x 16 c C C C C There are C’s in Part A A (continued on next page) (continued on next page) As students feel comfortable, encourage them to continue the problems on their own If they finish the models you’ve created, ask them to solve for an unknown on the challenge problems provided Reserve time at the end of the session for volunteers to prove their thinking © The Math Learning Center Bridges in Mathematics Grade Supplement • A10.3 Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Activity Multiplying Fractions on the Geoboard cont INDEPENDENT WORKSHEET See Pizza Party Planning on pages A10.17 and A10.18 for more practice with multiplying fractions by a whole number in the context of word problems A10.4 • Bridges in Mathematics Grade Supplement © The Math Learning Center Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Blackline Run a class set NAME DATE Fractions on a Geoboard Multiplying Fractions In the illustration below, the whole geoboard has an area of A B C D E Look at the visual models below and think about what you notice about the relationships between the regions on the geoboard How could you express your observations using multiplication? Explain your thinking in words and record your observations in a multiplication equation Visual Model Explanation Multiplication Equation a B C C b D D D D D D D D A c C C C C A (continued on next page) © The Math Learning Center Bridges in Mathematics Grade Supplement • A10.5 Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Blackline Run a class set NAME DATE Fractions on a Geoboard Multiplying Fractions cont Visual Model Explanation Multiplication Equation d E E E E C e B B f C C C C C C g D D D D (continued on next page) A10.6 • Bridges in Mathematics Grade Supplement © The Math Learning Center Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Blackline Run a class set NAME DATE Fractions on a Geoboard Multiplying Fractions cont CHALLENGE Create a sketch that could be used to solve the following unknowns: Problem a Labeled Sketch × _ = b × = 16 c × = d × = © The Math Learning Center Answer Bridges in Mathematics Grade Supplement • A10.7 A10.8 • Bridges in Mathematics Grade Supplement © The Math Learning Center Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Activity Quick Sketches & Fraction Arrays cont Teacher Turn to your partner and discuss how I could use this array to help me find 1/4 × Who would like to volunteer an idea? Jodi You can just cut the rectangle into fourths Teacher Can you show me what you mean? Jodi Sure (Points on grid.) You can just draw a line to divide the length of your rectangle in half That shows the two halves or 1/2 of Then split each half to make fourths Teacher I’m going to add the lines that Jodi mentioned to our grid In my drawing, where you see /4 × 3? Take a moment to tell your partner where you see the 1/4 × array Who would like to share? Megan I see it across the top row The top row just shows 1/4, so you could write 1/4 next to the top row Teacher I’m going to erase the 1, label our array with 1/4, and shade in the top of my drawing to show the problem we’re discussing Have students pair-share how they see the area pieces to determine the product of 1/4 × Is it the same as × 1/4 ? Can they prove it? Students may notice that they can multiply straight across Record their thoughts Joe I know you can put any number over and it’s still the same number over 1, that’s like just So I put over and multiplied straight across 3/1 × 1/4= 3/4 Liz Hey, that’s cool I wonder if it works every time? David You don’t even need the under the You can just look at × 1/4 and in your head go (3 × 1)/4 Write another multiplication combination on the board, /3 × 5, and repeat the process This time, begin by having students make and share estimates, first in pairs and then as a class Joy I think the answer is going to be more than one Like somewhere between 1/2 and /2 of would be 1/2 and I know /3 of is more than that A10.10 • Bridges in Mathematics Grade Supplement © The Math Learning Center Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Activity Quick Sketches & Fraction Arrays cont Ask students to determine the closest whole numbers for /3 and Have students draw a × rectangle, divide it into fractional parts, label the drawing, and shade in the area If students find /3 confusing, you may want to first label each row with 1/3, or thirds, then erase and label the top two rows with /3 Again, have students pair-share to determine the area of /3 × Then work with student input to label the sub-regions, recording the product and one or more equations to show how students figured it out • Students may look for groups of fractions that equal 1: 1 3 + 33 + 3 + 31 = 31 + + + 31 = 31 • Students may observe that each column shows /3 of whole: 3 3 3 3 + × = 103 + 23 + 23 + 23 = 103 ã They may also notice that /3 ì is the same as 1/3 × 10 3 3 3 3 3 × = (2 × 5) × or 10 × 31 3 ã Or ì /3 could be understood as times groups of 1/3 » n × (a /b) = (n × a)/b » In this case, × /3 = (5 × 2)/3 If students need additional modeling, another problem together: ì 1/3 Additional Resource ã On The Math Learning Center website, visit “A Play on Numbers” by Eugene Maier and read Scene 12: Multiplying/Dividing Fractions http://www.mathlearningcenter.org/media/gene/play-onnumbers/PON_S12_01.asp © The Math Learning Center Bridges in Mathematics Grade Supplement • A10.11 Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Activity Quick Sketches & Fraction Arrays cont INDEPENDENT WORKSHEET See Brownie Dessert on pagesA10.19 and A10.20 for more practice with equivalent fractions and word problems A10.12 • Bridges in Mathematics Grade Supplement © The Math Learning Center Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Set A10 H Activity ACTIVITY Fraction Word Problems & Arrays Overview You’ll need: Today’s session extends work using array models to multiply a whole number by a fraction Students sketch, discuss, and solve four story problems H pens and paper to record on the display Skills & Concepts: H Write an equation showing a fraction a/b as the sum of a number of the unit fraction 1/b (CCSS 4.NF.3) H Write an equation to show a fraction as the sum of other fractions with the same denominator (CCSS 4.NF.3b) H Pencil/paper for each student H Story Problems: Multiplying Fractions page A10.16, run a class set plus one for display Note When you represent the symbolic form for a fraction, please use a horizontal bar H Add fractions with like denominators H Demonstrate an understanding that a fraction a/b is a multiple of the unit fraction 1/b (CCSS 4.NF.4a) H Write an equation showing that a fraction a/b is the product of a × 1/b (CCSS 4.NF.4a) H Multiply a fraction by a whole number (CCSS 4.NF.4b) H Solve word problems in which fractions are multiplied by a whole number H Use a visual fraction area model to understand multiplication of fractions by a whole number Instructions for Fraction Word Problems & Arrays Give each student a copy of Story Problems: Multiplying Fractions while you display the same Teacher Master Read the first problem together, taking time to discuss and record students’ thoughts about what the problem is asking them to Pair-share estimates and then ask volunteers to explain their reasoning with the class Encourage students to consider benchmark fractions DATE Story Problems Multiplying Fractions Estimate then draw an array to show your work Problem The principal hands out ice cream bars and popsicles at field day The first class finished two bags of ice cream bars Only 14 of that amount of ice cream bars were needed for the second class How much ice cream did the second class eat? I have dogs This week each dog © The Math Learning Center Estimate Array and Answer Set A10 Number & Operations: Multiplying Whole Numbers by Fractions NAME Bridges in Mathematics Grade Supplement • A10.13 Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Activity Fraction Word Problems & Arrays cont Teacher When you estimate, sometimes it’s helpful to think about benchmarks or numbers that are easy to imagine In this problem I’m wondering if the answer will be closer to zero, one half, or one whole? Scott It must be close to zero 1/4 of bags has to be a really small number Jacque It’s small, but I don’t think it’s that small 1/4 of bag is 1/4 So 1/4 of bags has to be more My estimate would be closer to one half Next, work with the class to sketch an array to represent the problem 1/4 × Ask students to determine the closest whole numbers Have them draw a × rectangle, and coach you to divide it into fractional parts, as they sketch the same model Label the drawing, and shade in the area After students have had a chance to sketch the array, work with the class to connect to the original problem What the two dimensions represent? What does the area represent? Then ask students to find the area of the array they’ve drawn Have students pair-share their thoughts before a volunteer shares with the entire class Record multiplication equations to match student strategies Elsie I saw groups of 1/4 So I just added 1/4 and 1/4 That’s /4 Jeb I did the same thing But I said it equals 1/2 Because /4 and 1/2 are equivalent Sarah I just multiplied I know that you can put any number over a and it’s still the same number So I said /1 × 1/4 and just multiplied them across That’s /4 DATE Estimate then draw an array to show your work Problem Estimate Array and Answer The principal hands out ice cream bars and popsicles at field day The first class finished two bags of ice cream bars Only 14 of that amount of ice cream bars were needed for the second class How much ice cream did the second class eat? 2 Set A10 Number & Operations: Multiplying Whole Numbers by Fraction NAME Story Problems Multiplying Fractions • How are we seeing the area of this rectangle? » 1⁄4 + 1⁄4 = ⁄4 » 1⁄4 + 1⁄4 = 1⁄2 » 1⁄2 × 14 = ⁄4 With student input, label the area of each rectangle A10.14 • Bridges in Mathematics Grade Supplement 4 © The Math Learning Center Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Activity Fraction Word Problems & Arrays cont Repeat this process with the three other problems on the projector as students complete the problems on their own sheets INDEPENDENT WORKSHEET See Multiplying Fractions with Arrays on page A10.21 for more practice with estimating and sketching arrays for multiplying fractions by a whole number © The Math Learning Center Bridges in Mathematics Grade Supplement • A10.15 DATE Story Problems Multiplying Fractions Estimate then draw an array to show your work Problem Estimate The principal hands out ice cream bars and popsicles at field day The first class finished two bags of ice cream bars Only 14 of that amount of ice cream bars were needed for the second class How much ice cream did the second class eat? I have dogs This week each dog gets 18 of a bag of dog treats How much of the bag is eaten? The cookie recipe calls for cup of sugar I need to make a double batch of cookies How much sugar I need? © The Math Learning Center If everyone wants a pound hamburger, and we have six people in our family, how many pounds of hamburger we need? Array and Answer Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Blackline Run a class set, plus for display A10.16 • Bridges in Mathematics Grade Supplement NAME Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Blackline For use anytime after Supplement Set A10, Activity NAME DATE Set A10 H Independent Worksheet INDEPENDENT WORKSHEET Pizza Party Planning A fourth grade class won a pizza party for contributing the most paper in the recycling contest at school Medium pizzas were cut into slices and large pizzas were cut into 12 slices Mariah ate slices of a large pizza What fraction of the pizza did she eat? Draw a sketch to show your thinking Carlos said that Mariah ate of a pizza Explain why you agree or disagree Mariah’s table group seats four students Each student ate slices of a large pizza Write a multiplication equation that shows what fraction of a pizza was eaten (continued on next page) © The Math Learning Center Bridges in Mathematics Grade Supplement • A10.17 Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Blackline NAME DATE Independent Worksheet Pizza Party Planning (cont.) Tony ate slices of a medium pizza and his friend Conner ate slices of the same pizza a Write two equivalent fractions to tell how much pizza Conner ate b Write an equation to show what fraction of the pizza they ate together Lionel’s table group drank 1 liters of juice with their pizza How many milliliters did they drink? A10.18 • Bridges in Mathematics Grade Supplement © The Math Learning Center Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Blackline For use anytime after Supplement Set A10, Activity NAME DATE Set A10 H Independent Worksheet INDEPENDENT WORKSHEET Brownie Dessert A fourth grade class earned a brownie dessert party for the highest attendance in one grading period Smaller pans of brownies were cut into pieces and large pans were cut into 16 pieces Tori ate pieces of brownie from the smaller pan What fraction of the small pan of brownies did she eat? Draw a sketch to show your thinking Jorge ate one piece more than Tori Write two equivalent fractions to show how much Jorge ate Ford’s table group seats five students Each student ate pieces from the large brownie pan Write an equation that shows what fraction of the large pan of brownies was eaten April ate piece from the large pan and her friend Karissa ate slices from the same pan a Write two fractions to tell how much of the large brownie pan Karissa ate (continued on next page) © The Math Learning Center Bridges in Mathematics Grade Supplement • A10.19 Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Blackline NAME DATE Independent Worksheet Brownie Dessert (cont.) b What fraction of the large brownie pan did they eat together? Frederick had of the brownies from the large pan His brother said that he ate of the pan of brownies Explain why you agree or disagree CHALLENGE In a 12-egg carton, equals eggs Use the grids below to help you imagine and draw cartons where: a c is eggs b is 25 eggs How did you decide on the sizes of the cartons for a and b? A10.20 • Bridges in Mathematics Grade Supplement © The Math Learning Center Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Blackline For use anytime after Supplement Set A10, Activity NAME DATE Set A10 H Independent Worksheet INDEPENDENT WORKSHEET Multiplying Fractions with Arrays Sketch each array below Then write a multiplication equation to show how you found the product Sketch This Array a b ×2 c ×5 d × 10 Equation ×3 Write a story problem to match one of the combinations above © The Math Learning Center Bridges in Mathematics Grade Supplement • A10.21 A10.22 • Bridges in Mathematics Grade Supplement © The Math Learning Center Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Blackline For use anytime after Supplement Set A10, Activity NAME DATE Set A10 H Independent Worksheet INDEPENDENT WORKSHEET Company’s Coming! Jamie has three people in her family For three people, they make a recipe for Cranberry Sauce that calls for the following ingredients: • (12 ounce) bag of fresh cranberries • 34 cup orange juice • 23 cup brown sugar • cup white sugar a Jamie’s grandparents are joining them for dinner Double Jamie’s recipe for Cranberry Sauce Show your work and the new ingredient list below b Jamie’s aunt, uncle and cousin are now coming for dinner as well Triple the original recipe Show your work and the new ingredient list below (continued on next page) © The Math Learning Center Bridges in Mathematics Grade Supplement • A10.23 Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Blackline NAME DATE Independent Worksheet Company’s Coming! (cont.) Jamie also needs to plan for the meal The following portions are for one person To serve 8, how much of each food is needed? Write an equation to show your thinking Note: remember to label the units in your equation Food a Vegetables b Turkey c Serving cup ounces Gravy cup d Potatoes cup e Stuffing cup f cup cup pie Green Salad g Cranberry Show Your Work and Write an Equation for Servings Sauce h Apple Pie A10.24 • Bridges in Mathematics Grade Supplement © The Math Learning Center ... Grade Supplement • A10. 7 A10. 8 • Bridges in Mathematics Grade Supplement © The Math Learning Center Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Set A10 H Activity ACTIVITY... Supplement • A10. 11 Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Activity Quick Sketches & Fraction Arrays cont INDEPENDENT WORKSHEET See Brownie Dessert on pagesA10.19 and A10. 20... Supplement • A10. 3 Set A10 Number & Operations: Multiplying Whole Numbers by Fractions Activity Multiplying Fractions on the Geoboard cont INDEPENDENT WORKSHEET See Pizza Party Planning on pages A10. 17