Multiplication Arrays


 Lee Ellis
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1 Multiplication Arrays Objective To provide opportunities to use arrays, multiplication/ division diagrams, and number models to represent multiplication number stories. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Use basic facts to solve multiplication number stories. [Operations and Computation Goal 3] Use arrays to model multiplication. [Operations and Computation Goal ] Write number sentences to model number stories. [Patterns, Functions, and Algebra Goal ] Model the turnaround rule for multiplication (Commutative Property of Multiplication) using an array model. [Patterns, Functions, and Algebra Goal ] Key Activities Children use arrays to represent multiples of equal groups. Children solve multiplication number stories using arrays, multiplication/ division diagrams, and number models. Ongoing Assessment: Recognizing Student Achievement Use journal page 8. [Operations and Computation Goal ] Key Vocabulary array factor product Materials Math Journal, p. 8 Home Link transparency of Math Masters, pp. 0 and 9 (optional) per person: 0 counters slate number grid calendar number line (optional) calculator (optional) Measuring Perimeters of Polygons Math Journal, p. 8 ruler Children measure perimeters of polygons and draw figures with given perimeters. Math Boxes Math Journal, p. 83 Children practice and maintain skills through Math Box problems. Home Link Math Masters, p. 87 Children practice and maintain skills through Home Link activities. READINESS Building Arrays Math Masters, pp. 88 and centimeter cubes or other counters sixsided die Children build arrays. ENRICHMENT Exploring Square Numbers Math Masters, pp. 89 and per group: centimeter cubes or pennies (or both), tape Children explore square arrays. ELL SUPPORT Building a Math Word Bank Differentiation Handbook, p. 3 Children add the term array to their Math Word Banks. Advance Preparation Gather a large supply of pennies or other counters. Choose a place for an Arrays Exhibit. See Home Link for details. Display the Guide to Solving Number Stories on Math Masters, page 0 and a multiplication/division diagram. Teacher s Reference Manual, Grades 3 pp Unit Multiplication and Division
2 Getting Started Mathematical Practices SMP, SMP, SMP3, SMP, SMP5, SMP, SMP7 Content Standards 3.OA., 3.OA.3, 3.OA., 3.OA.5, 3.OA.7, 3.MD.7a, 3.MD.7b, 3.MD.8 Mental Math and Reflexes Have children count chorally. They may use number grids, number lines, calculators, or calendars as needed. Skip count forward and backward by s, 5s, and 0s. Skip count forward and backward by s. Skip count forward and backward by 8s. Math Message There are trombone players in a big parade. Use counters to represent trombone players. Arrange the counters to show them in equal rows. Home Link FollowUp Ask several children to share the multiples of equal groups they found. They will share some of their stories later in this lesson. Teaching the Lesson Math Message FollowUp WHOLECLASS Have a volunteer draw an array on the board showing how he or she arranged the counters. Use circles or Xs to represent trombone players. Ask others to draw different arrangements on the board. Explain that the pictures show the number of trombone players in arrays. An array is a group of objects arranged in rows and columns. Each row has the same number of objects, and each column has the same number of objects. Arrays provide a way to model multiplication. Possible arrays: by, by, 3by8, 8by3, by, by, by, and by rows per row = =. Draw an array on the board representing rows of trombone players with players in each row.. Display a multiplication/division diagram, and have a volunteer fill in the known quantities. 3. Write in the bottom right box.. Call attention to the fact that the numbers do not have the same units or labels, as they do in addition and subtraction stories. 5. Ask a volunteer to write a number model for this array: =.. Next, draw an array to represent rows of trombone players with players in each row. rows = = per row Lesson 9
3 Repeat Steps through 5: Fill in a multiplication/division diagram and write a number model. Compare the two arrays, emphasizing that although the total number of players is the same in each array, the number of rows and the number of players per row are reversed. Remind children of the turnaround rule for multiplication (Commutative Property of Multiplication), which states that two numbers can be multiplied in either order without changing the product: = Solving Multiplication Number Stories Using Arrays (Math Masters, p. 9) PROBLEM SOLVING WHOLECLASS Adjusting the Activity Hold up two egg cartons to provide a visual for eggs in cartons or dozen eggs. AUDITORY KINESTHETIC TACTILE VISUAL NOTE A factor is a number that is multiplied by another number. A product is the result of multiplying factors. It is beneficial to use these words when discussing multiplication number stories, but do not insist that children use them. Teaching Aid Master Multiplication/Division Diagram per per per ELL Children draw arrays on their slates to solve simple multiplication number stories. Pose multiplication number stories in which arrays are obvious, such as the following. boxes of crayons, 8 crayons per box shoe boxes, pair of shoes per box 5 rows of chairs, 3 chairs per row 5 shelves of books, 8 books per shelf cartons of eggs, eggs per carton For each story:. Write a brief summary of the story on the board.. Fill in a multiplication/division diagram, going over the number of rows and the number of items per row (the factors) and the total number of things (the product) as children tell you what information to enter into the diagram. To support English language learners, distinguish between the common meaning and mathematical meaning of product. 3. Ask a volunteer to write a number model for the story on the board while the other children write number models on their slates. Children may use either a? or a letter variable to represent the unknown quantity.. Have children draw arrays on their slates or arrange counters on their desks to model the story. As they do this, have a volunteer draw an array on the board or arrange counters on the overhead projector. 5. Have children solve the problem and share their solution strategies. Remind them to include the unit in the answer.. Ask children to discuss the reasonableness of their answers and whether their answers make the number model true. Finally, have children write summary number models on their slates for the story. Math Masters, p Unit Multiplication and Division
4 Representing Multiplication Situations with Arrays (Math Masters, p. 9) WHOLECLASS Have children share stories from Home Link  about multiples of equal groups. Follow steps 5 from the previous activity. Make the point that arrays can be used to represent multiplication situations, even when the reallife groups do not look like arrays. Arrays are easy to draw, and they simplify problem solving. Solving More Multiplication Number Stories (Math Journal, p. 8; Math Masters, p. 0) PARTNER PROBLEM SOLVING Partners or individuals make arrays and use diagrams to solve multiplication number stories. Children may use counters if they wish. Circulate and ask guiding questions such as those on the Guide to Solving Number Stories (Math Masters, page 0). When reviewing the answers to the journal page, ask followup questions so children can practice more difficult multiplication facts. For example: Suppose Mrs. Kwan has 7 boxes of markers. How many markers would she have now? 5 markers If Monica had 0 dolls, how many shelves would she need? 0 shelves Date Time More Multiplication Number Stories Fill in the multiplication/division diagram. Write a number model. Make an array with counters. Mark the dots to show the array. Find the answer. Write the unit with your answer.. Mrs. Kwan has 3 boxes of scented markers. Each box has 8 markers. How many markers does she have? markers markers boxes per box 3 8? Number model: 3 8 =? Answer: markers. Monica keeps her doll collection in a case with 5 shelves. On each shelf there are dolls. How many dolls are in Monica s collection? dolls dolls shelves per shelf 5? Number model: 5 =? Answer: 3. During the summer Jack mows lawns. He can mow lawns per day. How many lawns can he mow in 7 days? days lawns per day lawns 7? 30 dolls Number model: 7 =? Answer: 8 lawns Math Journal, p _EMCS_S_SMJ_G3_U0_57353.indd 8 Student Page (unit) (unit) (unit) // 9: AM Ongoing Assessment: Recognizing Student Achievement Journal Page 8 Use journal page 8 to assess whether children are making progress toward using arrays and multiples of equal groups to demonstrate the meaning of multiplication. Children are making adequate progress if they are able to draw arrays and find the answer for Problems,, and 3. Some children may be able to fill in the diagrams and write the number models. [Operations and Computation Goal ] Date Time Measuring Perimeter Measure the perimeter of each shape in inches... Student Page 50 5 Perimeter: inches Perimeter: 7 inches Try This Perimeter: 0 inches 5. Draw any shape with a perimeter of 0 units. Perimeter: 3 3_ inches Sample answer: What is the area of your shape? Sample answer: square units Math Journal, p _EMCS_S_SMJ_G3_U0_57353.indd 8 // 9: AM Lesson 5
5 Date Math Boxes. Complete the bar graph. Points Point Totals A B C D Players 3. Solve. Make a ballpark estimate to check that the answer makes sense. 857 = Player A scores points. Player B scores 8 points. Player C scores 3 points. Player D scores 9 points. estimate: Sample answer: = Fill in the empty frames. 3 Unit 3 5 Student Page 5 Time. 0 packs of gum on the shelf in the candy store. 8 sticks of gum per pack. How many sticks of gum in all? packs gum per of gum pack 0 8? Answer: 80 sticks sticks of. Solve. 3 5 = 5 3 nickels = 5 0 = 5 0 = nickels.,798 Which digit is in the tens place? 9 Which digit is in the hundreds place? 7 Which digit is in the ones place? 8 sticks Which digit is in the thousands place? Ongoing Learning & Practice Measuring Perimeters of Polygons (Math Journal, p. 8) Journal page 8 reviews content covered in Unit 3. Problems provide children with practice measuring line segments. Children measure the perimeters of polygons to the nearest _ inch. Problem 5 involves children drawing a figure with a given perimeter and then counting unit squares to find the area. Math Boxes (Math Journal, p. 83) Math Journal, p _EMCS_S_SMJ_G3_U0_57353.indd 83 // 0: AM Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson . The skill in Problem previews Unit 5 content. Writing/Reasoning Have children write an answer to the following: Why are estimates helpful in Problem 3? Sample answer: Estimates help you know whether your answer makes sense. Home Link (Math Masters, p. 87) HOME LINK Family Note Arrays Home Link Master Your child is learning how to represent multiplication problems using pictures called arrays. An array is a group of items arranged in equal rows and equal columns. Help your child use counters, such as pennies or macaroni, to build the array in each problem. Your child should record each solution on the dots next to the problem. 5 Please return this Home Link to school tomorrow. For the next few weeks, look for pictures of items arranged in equal rows and columns, or arrays. Look in newspapers or magazines. Have people in your family help you. Explain that your class is making an Arrays Exhibit. This is a 5by array. There are 5 rows. There are dots in each row. There are 30 dots, since 5 = 30. Home Connection Children use arrays to solve multiplication stories. They find pictures from newspapers, magazines, and other sources that show arrays of items. You may want to use these pictures to build an Arrays Exhibit. NOTE Some teachers include a small section in their Arrays Exhibit titled Almost an Array. This section may include children s contributions that are close but not quite correct. It is important to discuss why such contributions are considered almost an array but not a true array. Make an array with counters. Mark the dots to show the array.. rows with counters per row a by array counters. 3 rows with 8 counters per row a 3 8 array counters 3. rows with counters per row a array counters Practice Write these problems on the back of this page. Solve. Show your work = = = 8 Unit Math Masters, p _EMCS_B_MM_G3_U0_57957.indd 87 /3/ 0: AM 5 Unit Multiplication and Division
6 3 Differentiation Options READINESS Building Arrays (Math Masters, pp. 88 and ) 5 5 Min Materials Teaching Master Building Arrays cm cubes or other counters sixsided die centimeter grid paper (Math Masters, p. ). Roll the die times. The first number tells how many rows to put in the array. The second number tells how many counters to put in each row. Example You roll a first and then a 5. You make this figure. To provide children with experience forming arrays, children build arrays with counters, following directions on Math Masters, page 88. They record the arrays on centimeter grid paper, Math Masters, page, and fill in the Array Table on page 88.. Draw the arrays you make on centimeter grid paper. 3. Then, fill in the Array Table below. How many How many cubes in each How many rows? row? cubes? ENRICHMENT Exploring Square Numbers (Math Masters, pp. 89 and ) SMALLGROUP 5 30 Min 3 5 Algebraic Thinking To further explore arrays, have children build square arrays with counters for as many numbers as possible, beginning with. They record their arrays on centimeter grid paper (Math Masters, page ) and write a number model under each array. As children build arrays for larger products, they might need to tape sheets of grid paper together. Have children arrange the number models and products for each array in order from smallest to largest. Ask: What patterns do you see? Sample answer: The differences between consecutive products are odd numbers, and they increase by. Math Masters, p. 88 ELL SUPPORT SMALLGROUP Building a Math Word Bank (Differentiation Handbook, p. 3) 5 5 Min To provide language support for multiplication, have children use the Word Bank template found on Differentiation Handbook, page 3. Ask children to write the word array, draw a picture representing the word, and write other related words. See the Differentiation Handbook for more information. Square Products Work in a small group. Teaching Master Materials centimeter grid paper (Math Masters, p. ) centimeter cubes or pennies (or both) tape Directions. Each person chooses a different number from to 0.. Build an array that shows your number multiplied by itself. Use centimeter cubes or pennies. 3. Draw each array on centimeter grid paper. Write each number model under each array Continue to build arrays with other numbers. Draw the arrays on grid paper. You may need to tape pieces of grid paper together for the larger arrays. 5. Look at the arrays you made. Why do you think the products of the number models you wrote are called square products? Sample answer: All of the arrays are squares. Math Masters, p. 89 Lesson 53
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