Exploring Arrays, Coins, and Division

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1 Explorations Exploring Arrays, Coins, and Division Objectives To develop readiness for multiplication; to guide children in finding coin combinations equivalent to $1.00; and to explore one meaning of division. 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 Key Concepts and Skills Count dots in an array. [Number and Numeration Goal 1] Create equal-size groupings. [Operations and Computation Goal ] Use geoboards to create arrays. [Operations and Computation Goal ] Create complements of $1.00 using nickels, dimes, and quarters. Key Activities Exploration A: Children make arrays on geoboards, record these arrays on dot paper, and sort arrays having the same number of dots into groups. Exploration B: Children make and record different combinations of nickels, dimes, and quarters that are equivalent to $1.00. Exploration C: Children determine how many groups of n objects can be made from a specified number of objects and how many objects are left over. Ongoing Assessment: Recognizing Student Achievement Use an Exit Slip (Math Masters, page 15). Materials Home Link 6 5 slate Exploration A: Per child (except where noted): Math Masters, p. 17; p. 17 or 17 geoboard rubber band overhead geoboard (optional) per group: scissors, large sheet of paper (optional), glue or paste (optional) Exploration B: Per child: Math Masters, pp. 175 and 15 0 nickels, 10 dimes, quarters half-dollars (optional) paper Exploration C: Per child (except where noted): Math Journal 1, p. 16 Math Masters, p. 176 per group: container with about 50 pennies or other counters, 1 six-sided die Ongoing Learning & Practice Making a Picture Graph Math Journal 1, pp. 16A and 16B Math Boxes Math Journal 1, p. 17 Home Link Math Masters, p. 177 Differentiation Options ENRICHMENT Solving Dollar Riddles Math Masters, p. 178 tool-kit coins EXTRA PRACTICE Minute Math + Minute Math +, pp Advance Preparation Plan to spend most of your time working on Exploration A with children. Math Masters, pages 17, 175, and 176 give directions for the Explorations. Teacher s Reference Manual, Grades 1 pp Lesson 07

2 Getting Started Mental Math and Reflexes Write subtraction problems like the following on the board. Have children write ballpark estimates and the number models they used to make them on their slates = = 5; or 0-0 = = 10 Math Message How many dots? Mathematical Practices SMP1, SMP, SMP5, SMP6, SMP7 Content Standards.OA.,.MD.8,.MD.10 Home Link 6 5 Follow-Up As children share their answers, circulate and note how they crossed out (subtracted) cubes. Some children will not cross out the cubes they are subtracting in the simplest way. For example, in Problem, a child may cross out 5 cubes by crossing out 5 from each of the first longs and 10 from the final long leaving 5, 5, 5, 0, and cubes. Because 5 is tens and 5 ones, it can be represented by longs and 5 cubes. The simplest approach is to cross out longs, then the cubes, and finally more cubes on a remaining long. 1 Teaching the Lesson How many dots in each row? Geoboard Arrays Materials geoboard dot paper for each person geoboard for each person rubber band for each person scissors for the group glue or paste for the group (optional) large sheet of paper for the group (optional) Work by yourself to complete Steps Use one rubber band to make a rectangle on your geoboard. The pegs inside and the pegs that touch the rubber band make an array.. Draw your array on the geoboard dot paper.. Write about your array at the bottom of the geoboard dot paper. Tell how many rows are in your rectangle, how many dots are in each row, and how many dots in all are in your rectangle.. Make more arrays all different. Follow steps two and three. How many dots in all? In Exploration A, children use either a 5 5 geoboard (Math Masters, page 17), or a 7 7 geoboard (Math Masters, page 17.) There are rows of 5 pegs. 10 pegs are in the array. 5. Cut apart the dot-paper records of your arrays. Work with your group to complete Step Sort your group s arrays into piles that have the same number of dots. You might want to use the arrays in each pile to make a display about that number. Math Message Follow-Up Ask: How many rows of dots are there? How many dots are in each row? 5 How many dots are there in all? 10 WHOLE-CLASS Use an overhead geoboard to show children how to use a rubber band to enclose 10 pegs in a 5 rectangle or demonstrate this on a regular geoboard and pass it around the classroom. Explain that in one of the Explorations in this lesson, children will use rubber bands to form rectangles and then count the number of pegs in the enclosed array including the ones touching the rubber bands. Exploration A: Making Geoboard Arrays (Math Masters, p. 17; and p. 17 or 17) PARTNER Directions for making rectangles on a geoboard are found on Math Masters, page 17. Before proceeding, check the size of your class s geoboards. If the geoboards in your classroom are 5 5, use Math Masters, page 17. If the geoboards are 7 7, use Math Masters, page 17. Math Masters, p Unit 6 Whole-Number Operations and Number Stories

3 Exploration B: Making a Dollar (Math Masters, p. 175) Before children begin, ask them to think about an organized way to determine the different groups of coins equivalent to $1.00. Children are given a hint: Find ways of using quarters and other coins. Children plan how to find the coin combinations and then record the groups of coins using Â, Í, and. Making a Dollar Work together in a small group. Materials 0 nickels 10 dimes quarters paper and pencil Directions 1. Use the coins to find as many different ways as you can to make $ Before you begin, THINK about how to do this. Hint: First, make a dollar using quarters and some other coins.. Plan how you will record the different ways to make $ On a sheet of paper, record the different ways you find to make $1.00. Use Â, Í, and to show the coins. Follow-Up How many ways did you find to make $1.00? Check with other groups to see if they thought of any ways that your group didn t find. Did you have a plan to find all the combinations? Compare your plan with the plan used by another group. Math Masters, p. 175 Coin combinations Adjusting the Activity Add two half-dollars to the set of coins. How Many Children Get n Things? Follow the directions on Math Masters, page 176 to fill in the table. What is the How many counters How many How many total number are in each group? groups are counters are of counters? (Roll a die to find out.) there? left over? A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L Ongoing Assessment: Recognizing Student Achievement Exit Slip Use an Exit Slip (Math Masters, page 15) to assess children s ability to make bill and coin exchanges. Have children record as many combinations for $1.00 as they can. Children are making adequate progress if they can draw four or more combinations. Some children may be able to find all possible combinations. Exploration C: Finding How Many Children Get n Things (Math Journal 1, p. 16; Math Masters, p. 176) Algebraic Thinking Using instructions from Math Masters, page 176, children make as many equal piles as possible. Then, on journal page 16, they record the total number of counters, the number of counters in each group, and the total number of groups. They also record how many counters are left over; that is, how many are not in a full group. NOTE You may want to vary Step 1 of this activity by having children estimate the number of counters that are in the pile before they begin counting. Have them compare their estimates to the actual number of counters. Children record totals in Exploration C on Math Journal, page 16. Materials How Many Children Get n Things? Math Journal 1, p. 16 (per person) one container with about 50 pennies or other counters (per group) 1 six-sided die (per group) Use counters to make up and solve problems like this one: Your group has been given crayons. Each person is to get 8 crayons. How many of you will get 8 crayons? Are there any crayons left over? Now make up your own problems. Follow these steps: 1. Each person takes a handful of counters. Put all the counters together in a pile. How many counters are in the pile? Count them and record the number on the journal page.. Make equal-size groups of counters. One person rolls the die. The number that lands faceup tells how many counters to put in each group. Record this number on the journal page.. Make as many groups as you can with the counters in the pile.. Record on the journal page how many groups you made. If any counters are left over, record that number, too. 5. Put the counters back in the container. Repeat Steps 1. Math Masters, p. 176 Lesson 09

4 Student Page Making a Picture Graph A zoo has a collection of birds. One month, they recorded the number of eggs the birds laid. The number of eggs a bird lays at one time is called a clutch. Use the information in the table below to create a picture graph. Type of American Canada Flamingo Mallard Toucan Bird Robin Goose Duck Number of Eggs in a Clutch Number of Eggs in a Clutch American Robin KEY: Bird Eggs Laid in 1 Month Canada Goose = 1 egg Math Journal 1, p. 16A Flamingo Mallard Duck Toucan Links to the Future In Lesson 5-1, children had experience with equal-sharing division situations. In this activity, children work with equal-grouping division situations. It isn t important that children be able to distinguish between the two meanings of division at this point. However, it is important that they have had informal experience with both meanings. In Lesson 6-10, children will begin to formalize their understanding of equal-sharing and equal-grouping. This is an early exposure to the concept of division. Demonstrating automaticity with all multiplication facts and proficiency with related division facts are Grade Goals. Ongoing Learning & Practice Making a Picture Graph (Math Journal 1, pp. 16A and 16B) ELL Children use data to make a picture graph and answer questions. To support English language learners, write the word clutch on the board and discuss the various meanings it has. Sometimes a person will clutch something to hold onto it. A bird clutch is the collection of eggs that a bird lays at one time. Math Boxes (Math Journal 1, p. 17) Making a Picture Graph continued Use the information from the picture graph on journal page 16A to solve the number stories. 1. The American Robin and Toucan are tree birds. What is the total number of eggs laid by the tree birds? Answer: 10 eggs Number model: 6 + = 10. The Canada Goose, Flamingo, and Mallard Duck are water birds. How many eggs did the water birds lay in total? Answer: eggs Number model: =. How many more eggs did the Mallard Duck lay than the American Robin? Answer: 6 eggs Number model: 1-6 = 6. Use the information from the graph to write your own number story. Have a partner solve the problem. Answer: Answers vary. Number model: Student Page Mixed Practice Math Boxes in this lesson are linked with Math Boxes in Lessons 6-8 and The skill in Problem 6 previews Unit 7 content has 6 9 Math Boxes hundreds tens ones. Share 1 dozen cookies equally among 5 children. Draw a picture. Each child gets cookies. There are cookies left over. 5. The temperature is F. F Student Page 5. Draw the line of symmetry Use counters to make a 5-by- array. Draw the array. How many counters in all? 10 counters 6. Fill in the pattern Math Journal 1, p. 16B Math Journal 1, p Unit 6 Whole-Number Operations and Number Stories

5 Home Link Master Writing/Reasoning Have children draw their answers to the following to extend Problem : Draw a shape that has one line of symmetry. Next, draw another shape that has more than one line of symmetry. Sample answers: An isosceles trapezoid; a circle or square Home Link (Math Masters, p. 177) Home Connection Children draw arrays with specified numbers of Xs. For each array, children give the number of rows and the number of Xs in each row. Differentiation Options HOME LINK How Many? Family Your child has been working with arrays rectangular arrangements of objects having the Note same number of objects in each row to develop readiness for multiplication. Because this is a readiness activity, children have not yet written number models for multiplication, such as 5 = 0. Your child will do this in later lessons in this unit. Please return this Home Link to school tomorrow. Sample answers: 1. Show someone at home this array. in each row? 5 in all? 0. Draw an array of Xs. in each row? 8. Draw an array of 16 Xs. in each row? 8. Draw a different array of Xs. in each row? 6 ENRICHMENT Solving Dollar Riddles (Math Masters, p. 178) PARTNER 15 0 Min Math Masters, p. 177 To apply children s understanding of coins and coin combinations, have them solve dollar riddles. Have children share strategies for solving some of the more difficult problems. EXTRA PRACTICE Minute Math Min To offer children more experience with problems including money, see the following pages in Minute Math+: pages Solving Dollar Riddles 1. To make a dollar, use all four types of coins and create a coin combination where there are two times as many of one type of coin as another. Sample answers: 10 pennies, 5 dimes,1 quarter, nickels; 10 nickels, 5 pennies, 1 quarter, dimes. To make a dollar, use all four types of coins. Use half as many of one type of coin. Sample answers: nickels, quarters, dimes, 10 pennies; quarters, 1 dime, 7 nickels, 5 pennies. To make a dollar, use only nickels and dimes and create a coin combination where one type of coin is used twice as much as the other type. Ten nickels and five dimes. Using only three types of coins, make a dollar with the least number of coins you could use. quarters, dimes, 1 nickel Using only three types of coins, make a dollar with the greatest number of coins you could use. 85 pennies, 1 dime, 1 nickel Try This Use pennies, nickels, dimes, and quarters. Make a combination that is worth one dollar where you have one of some kind of coin, double of another, double that of another, and some number of the last coin. Sample answers: dimes, quarters, 1 nickel, 5 pennies; nickels, quarters,1 dime, 0 pennies Math Masters, p. 178 Lesson 11

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