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Focus on Math Concepts Lesson 7 Part 1: Introduction CCSS 3.OA.D.9 Understand Patterns What are patterns? A pattern is something that repeats. Sometimes we see patterns in shapes or letters. Other times numbers make a pattern. Patterns can also be found when we use numbers to add, subtract, multiply, or divide. You can also see patterns in things around you. Look at the line of kids below. Think How can you describe a pattern? What do you add to get to the next number in the pattern? Telling about what repeats in a pattern is called the rule. The rule for the line of kids is boy, boy, girl. You can also use numbers to describe this pattern. 1 2 3 4 5 6 7 8 9 The numbers 3, 6, 9, . . . tell where the girls are in line. Because the numbers in the pattern do not repeat, you need to look for something that is done over and over to get from one number to the next number. 52 L7: Understand Patterns ©Curriculum Associates, LLC Copying is not permitted. Part 1: Introduction Lesson 7 Think How do you know what numbers come next in a pattern? You can use the rule to figure out what other numbers are in the pattern. To get from one number to the next in the pattern, you add 3. The pattern is continued in the chart below. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Putting the numbers in a chart helps me notice things I might not see if I just made a list of numbers. 24 25 26 You may also notice other things about the pattern that can help you figure out what numbers come next. The numbers form diagonals in the hundreds chart. You can use them to tell the next number in the pattern is 27. The numbers in this pattern also alternate: even, odd, even, odd. Since 27 is odd, you know that the number that comes after 27 will be even. Reflect 1 Write your own number pattern that has at least six numbers in it. Then, describe the rule and one other thing you notice about the pattern. L7: Understand Patterns ©Curriculum Associates, LLC Copying is not permitted. 53 Part 2: Guided Instruction Lesson 7 Explore It Use the information below to help you think about patterns in addition. Rick has a pack of 100 baseball cards and likes to sort them into 2 piles. He notices that when he has a pile of 20 cards, the other pile has 80 cards. When he has a pile of 30 cards, the other pile has 70. When he has a pile of 40 cards, the other pile has 60. Finally, when he has a pile of 50 cards, the other pile has 50 too. 2 Rick shaded 20 squares in the first grid to show how many baseball cards were in the first set of piles he made. Shade the rest of the grids to show the other sets of piles. 3 What do the shaded squares show in each grid? 4 What do the white squares show in each grid? 5 What happens to the number of shaded squares as you move from one grid to the next? 6 What happens to the number of white squares as you move from one grid to the next? 7 Describe the rule for this pattern. 54 L7: Understand Patterns ©Curriculum Associates, LLC Copying is not permitted. Part 2: Guided Instruction Lesson 7 Talk About It Solve the problems below as a group. 8 The grid on the left shows Rick’s piles of 30 and 70. The grid on the right shows Rick’s piles of 40 and 60. Shade the grid in the middle to show what happens if Rick put 35 in one pile. 9 Explain how Rick can use the pattern to find the number of cards in the other pile. 10 Describe what happens in addition patterns where the sum stays the same but you change the numbers you add together. Try It Another Way Work with your group to use the tables to show patterns with addends and sums. 11 Fill in the missing numbers. Addend Addend 20 30 12 Fill in the missing numbers. Sum Addend Addend Sum 100 100 100 100 100 5 8 10 16 20 20 25 25 25 25 25 80 65 60 50 L7: Understand Patterns ©Curriculum Associates, LLC Copying is not permitted. 55 Part 3: Guided Practice Lesson 7 Connect It 13 Explain: Izzy noticed a pattern in the addition table. She found a diagonal that had all 5s in it. Fill in the table below on the right to show the addends. 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 2 3 4 5 6 2 2 3 4 5 6 7 3 3 4 5 6 7 8 4 4 5 6 7 8 9 5 5 6 7 8 9 10 Addend 0 Addend 4 3 2 4 5 Sum 5 5 5 5 5 5 Explain why the 5s form a diagonal line. 14 Examine: Jace counted to 50 by fives. Annabel counted to 50 by tens. What numbers did both Jace and Annabel say? Explain why some of the numbers they said were the same. 15 Determine: Pat saw an odd number of birds on Monday and an even number of birds on Tuesday. Is the total number of birds he saw odd or even? Explain how you know this, even though you don’t know how many birds he saw. 56 L7: Understand Patterns ©Curriculum Associates, LLC Copying is not permitted. Part 4: Common Core Performance Task Lesson 7 Put It Together 16 Look at the multiplication table below. 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 2 0 2 4 6 8 3 0 6 9 12 4 0 4 12 20 28 36 5 0 5 15 25 35 45 6 0 6 12 18 24 7 0 7 14 21 28 8 0 9 0 18 15 18 21 36 35 18 27 48 54 49 40 48 9 24 63 64 36 45 54 63 81 AFill in the missing numbers. BDescribe a pattern you see in the table. CExplain why the pattern works the way it does. L7: Understand Patterns ©Curriculum Associates, LLC Copying is not permitted. 57 Focus on Math Concepts Lesson 7 (Student Book pages 52–57) Understand Patterns Lesson Objectives The Learning Progression •Use hundreds charts, addition tables, and multiplication tables to model addition or multiplication patterns and explain why patterns make sense. This lesson builds on students’ previous informal experiences with repeating words, shapes, or numbers. In grade 3 students develop understanding of what a pattern is and identify basic arithmetic patterns. •Use (informally) number properties to find and explain patterns. A major focus in this lesson is exploring patterns in an addition table. Students identify patterns in addends that make the same sum. When comparing all the facts, they notice that as the first addend increases by 1, the second addend in each fact decreases. •Use knowledge of even and odd numbers to find and explain patterns. Prerequisite Skills •Use addition, subtraction, multiplication, and division. •Complete an addition table and a multiplication table. •Recognize patterns in numbers. •Know the difference between even and odd numbers. Vocabulary Students also explore patterns in a multiplication table. A multiplication table has many patterns, so students should have multiple experiences over time using the table to find patterns. Some patterns help students find products for unknown multiplication facts. Students’ work with patterns in grade 3 provides a foundation for describing, generating and analyzing more complex patterns and eventually, examining relationships between ordered pairs, coordinate graphs, and studying proportional relationships and functions. pattern: a series of numbers or shapes that follow a rule to repeat or change Teacher Toolbox rule: a procedure that is followed to go from one number or shape to the next in a pattern Prerequisite Skills Review the following key terms. Ready Lessons even number: a number than can be divided into two equal groups Tools for Instruction odd number: a number that cannot be divided into two equal groups Interactive Tutorials Teacher-Toolbox.com 3.OA.D.9 ✓ ✓ CCSS Focus 3.OA.D.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends. STANDARDS FOR MATHEMATICAL PRACTICE: SMP 2, 4, 6, 7 (see page A9 for full text) L7: Understand Patterns ©Curriculum Associates, LLC Copying is not permitted. 59 Part 1: Introduction Lesson 7 At a Glance Students explore what a pattern is and how they can describe patterns. Focus on Math concepts Lesson 7 A pattern is something that repeats. Sometimes we see patterns in shapes or letters. Other times numbers make a pattern. Patterns can also be found when we use numbers to add, subtract, multiply, or divide. You can also see patterns in things around you. Look at the line of kids below. think How can you describe a pattern? You can also use numbers to describe this pattern. 1 60 2 3 4 5 6 7 8 9 The numbers 3, 6, 9, . . . tell where the girls are in line. Because the numbers in the pattern do not repeat, you need to look for something that is done over and over to get from one number to the next number. 52 •Explain that you can also attach numbers to the sound pattern, like they did for the girls and boys in line. Explore doing this with students. What do you add to get to the next number in the pattern? 3 Telling about what repeats in a pattern is called the rule. The rule for the line of kids is boy, boy, girl. •Read the Think question together. Explain that there can be more than one way to describe a pattern. If the students were numbered, then the numbers 3, 6, 9, etc. show where the girls are in the line. Emphasize that in order for shapes or numbers to be in a pattern, something must be seen or done to them over and over. Ask students for the next 3 numbers in the number pattern. [12, 15, 18] Have them explain how they found these numbers. •Explain to students that sounds can also be patterns if they repeat over and over. Use a drum stick or percussion instrument, such as a shaker to make a sound pattern (ex: tap….taptap…tap). Be sure to repeat the pattern at least 3 times so students can hear how the sounds repeat. Direct students to tap the same pattern on their desk. Ask them to describe the pattern (one tap, two taps, one tap and then it repeats). 3.oa.D.9 What are patterns? •Introduce the Question at the top of the page. To assess what students already know about patterns, ask them to use their own words to explain what patterns are and give examples. Use their ideas and examples to clarify what patterns are and are not. Direct students’ attention to the pattern shown on the top of the page and have them explain what is repeating. Experience and describe a sound pattern. ccss Understand Patterns Step By Step Concept Extension Part 1: introduction L7: Understand Patterns ©Curriculum Associates, LLC Copying is not permitted. Mathematical Discourse •How can you prove that something is a pattern? You must see the same numbers or shapes, hear the same sounds, feel the same things over and over. •Describe this pattern: beep, beep, bop, beep, beep, bop, beep, beep, bop. Can you attach numbers to this sound pattern? 1, 1, 2, 1, 1, 2 or 2 beeps, 1 bop (2, 1, 2, 1) •How do you know it’s a pattern? Students should recognize that the sounds repeat. L7: Understand Patterns ©Curriculum Associates, LLC Copying is not permitted. Part 1: Introduction Lesson 7 At a Glance Students explore ways to make number patterns visible by using a grid. Step By Step •Read the Think question together as a class. Direct students’ attention to the chart on the page. Explain that the chart is not complete, but that the rule for the pattern on the chart is “add 3.” •Instruct students to work in pairs to study the chart and discuss what they notice about the pattern that could help them figure out what number the next jump will show. Students may notice the difference between each diagonal number is 9, so they know that the next number will be 18 1 9, or 27. •Have students read and reply to the Reflect directive. You may ask students to work in groups to create the pattern and share. This allows you to assess student understanding and correct any misconceptions at this point in the lesson. •You may wish to give students practice circling number patterns, instead of highlighting them. Point out that when counting on a number chart, they may wish to count out loud. This can be a good strategy because some patterns may be easier for some people to hear than to see. STUDENT MISCONCEPTION ALERT: Some students may see 2 white boxes between each shaded box and describe the pattern incorrectly as a plus 2 pattern. Point out that the third box is shaded and you can count each box starting with 1 to reach number 3. Explain that you will have counted 3 boxes, so it is a plus 3 pattern. Mathematical Discourse •How could you prove that the chart shows a “plus 3” pattern? The third box is colored in for all the numbers on the chart. •Is there another way to describe the plus 3 pattern? Go by three, count by 3, or multiples of 3. 2 white boxes, 1 shaded box, the multiples are in diagonal lines. L7: Understand Patterns ©Curriculum Associates, LLC Copying is not permitted. Part 1: introduction Lesson 7 think How do you know what numbers come next in a pattern? You can use the rule to figure out what other numbers are in the pattern. To get from one number to the next in the pattern, you add 3. The pattern is continued in the chart below. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Putting the numbers in a chart helps me notice things I might not see if I just made a list of numbers. 24 25 26 You may also notice other things about the pattern that can help you figure out what numbers come next. The numbers form diagonals in the hundreds chart. You can use them to tell the next number in the pattern is 27. The numbers in this pattern also alternate: even, odd, even, odd. Since 27 is odd, you know that the number that comes after 27 will be even. reflect 1 Write your own number pattern that has at least six numbers in it. Then, describe the rule and one other thing you notice about the pattern. Possible answer: 0, 5, 10, 15, 20, 25. the rule is add 5. the ones digits have a pattern of 0, 5, 0, 5, 0, 5. L7: Understand Patterns ©Curriculum Associates, LLC Copying is not permitted. 53 Hands-On Activity Create visual patterns on a number chart. Materials: multiple copies of 1–100 number charts, highlighters or crayons. •Students benefit from many opportunities to create and describe a variety of plus and minus patterns on a number chart. •Ask students to highlight or shade a plus 2 pattern. On another chart have them highlight a plus 4 pattern. Give students opportunities to describe and discuss the patterns for plus 2 and plus 4. Then have them compare the two patterns. Ask: How are the patterns alike? Different? Have them highlight a plus 6 pattern and a plus 3 pattern on different charts. Ask them to describe any patterns they see. Ask: How are the patterns alike? Different? How do know it’s a pattern? •You may wish to create and make copies of a hundreds chart that begins with 100 and goes to 1. Students can use the chart to show subtraction patterns. 61 Part 2: Guided Instruction Lesson 7 At a Glance Part 2: guided instruction Students explore a different way of thinking about patterns. They see the pattern as a “doing” pattern in which 10 is taken from one pile and given to another pile. explore it use the information below to help you think about patterns in addition. Rick has a pack of 100 baseball cards and likes to sort them into 2 piles. Step By Step He notices that when he has a pile of 20 cards, the other pile has 80 cards. When he has a pile of 30 cards, the other pile has 70. When he has a pile •Work through the first four questions together. Explain that each grid shows how Rick split up the pack of 100 cards. On the board write “20 1 80 5 100.” Ask students how he split the cards up the next time. Underneath the first problem, write “30 1 70 5 100.” Writing the addends on the board gives students one more way to think about problem and the pattern. of 40 cards, the other pile has 60. Finally, when he has a pile of 50 cards, the other pile has 50 too. 2 Rick shaded 20 squares in the first grid to show how many baseball cards were in the first set of piles he made. Shade the rest of the grids to show the other sets of piles. 3 What do the shaded squares show in each grid? the number of cards in the first pile •Tell students that they will have time to work individually on the rest of the Explore It problems on this page and then share their responses in groups. •As students work individually, circulate among them. This is an opportunity to assess student understanding and address student misconceptions. Ask questions such as: What is happening to the shaded squares? As you compare all the grids, what is happening that repeats? How do you know? Lesson 7 4 What do the white squares show in each grid? the number of cards in the second pile 5 What happens to the number of shaded squares as you move from one grid to the next? the number of shaded squares gets bigger by ten. 6 What happens to the number of white squares as you move from one grid to the next? the number of white squares gets smaller by ten. 7 Describe the rule for this pattern. Possible answer: there are always 100 cards. every time you add some to the first pile, you have to take the same amount away from the second pile. 54 L7: Understand Patterns ©Curriculum Associates, LLC Copying is not permitted. •Take note of students who are still having difficulty and wait to see if their understanding progresses during the next part of the lesson. •As a class, share the answers to problems 5–7. Explain that a pattern can also be something that you do or that happens with numbers over and over. In this problem, as the number of shaded squares gets bigger, the number of white squares gets smaller. This idea may be new to some students. Point out that getting bigger (shaded squares) and getting smaller (white squares) are patterns. Concept Extension Help students see the pattern using number sentences. •Underneath the 30 1 70 on the board, ask students for the number sentence to write when Rick had a pile of 40 and 60. [40 1 60 5 100] Complete the chart to show all the ways Rick could have sorted his cards by groups of 10. Write the number sentences for each way. Point out that the two groups of cards always add up to 100. 62 Visual Model •Some students may need additional support to help them to see patterns in addends that equal the same number. Use tiles or strips to show the different ways to make 10. Then write the number sentence for each way: 1 1 9 5 10 2 1 8 5 10 3 1 7 5 10 etc. L7: Understand Patterns ©Curriculum Associates, LLC Copying is not permitted. Part 2: Guided Instruction Lesson 7 At a Glance Students extend their thinking about addition patterns where the sum stays the same, but the addends change in a certain way. Part 2: guided instruction Lesson 7 talk about it solve the problems below as a group. 8 The grid on the left shows Rick’s piles of 30 and 70. The grid on the right shows Step By Step Instruct students to continue to work in groups to answer problems 8210. Walk around to each group, listen to, and join in on discussions at different points. Use the Mathematical Discourse questions to help support or extend students’ thinking. •Ask groups to share their thinking about problems 8210. A possible answer for problem 9 is that Rick always has 100 cards. If he starts with a pile of 30, he knows the other pile must have 70 cards in it. So, if he has a pile with 35 cards (5 more), then he must have 5 less in the other pile (65). •Direct the group’s attention to Try It Another Way. Ask students to study the two tables and then ask a volunteer to explain how they work. Instruct groups to complete the tables. •Invite two groups to draw the table on the board and describe the patterns in the chart. Expect groups to use precise math language in their explanations using terms, such as addends, sum, rule, and pattern. Students should share that all addends in the each table equal the same sums and as one addend gets larger, the other addend gets smaller. SMP Tip: Students practice using precise mathematical language when explaining their ideas (SMP 6). L7: Understand Patterns ©Curriculum Associates, LLC Copying is not permitted. Rick’s piles of 40 and 60. Shade the grid in the middle to show what happens if Rick put 35 in one pile. 9 Explain how Rick can use the pattern to find the number of cards in the other pile. Possible answer: rick always has 100 cards. if he starts with the piles of 30 and 70, he has to add 5 to the 30 and take away 5 from the 70. this gets him piles of 35 and 65. 10 Describe what happens in addition patterns where the sum stays the same but you change the numbers you add together. Possible answer: When you make one of the numbers you add bigger, you must make the other number smaller by the same amount so that the sum will stay the same. try it another Way Work with your group to use the tables to show patterns with addends and sums. 11 Fill in the missing numbers. addend addend 20 30 sum addend addend sum 100 100 100 100 100 5 8 10 16 20 20 17 15 9 5 25 25 25 25 25 80 70 35 40 65 60 50 50 12 Fill in the missing numbers. L7: Understand Patterns ©Curriculum Associates, LLC Copying is not permitted. 55 Mathematical Discourse •How can using a chart help you to see patterns when you add numbers with the same sum? Students should recognize that charts often allow you to see several steps of a pattern all at once. 63 Part 3: Guided Practice Lesson 7 At a Glance Part 3: guided Practice Students demonstrate their understanding of using an addition table and charts to show addends with the same sum. They reason about the patterns they find. Lesson 7 connect it 13 explain: Izzy noticed a pattern in the addition table. She found a diagonal that had all 5s in it. Fill in the table below on the right to show the addends. Step By Step •Discuss each Connect It problem as a class using the discussion points outlined below. 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 2 3 4 5 6 2 2 3 4 5 6 7 3 3 4 5 6 7 8 4 4 5 6 7 8 9 5 5 6 7 8 9 10 addend 0 addend 1 2 3 4 3 2 4 5 1 0 5 sum 5 5 5 5 5 5 Explain why the 5s form a diagonal line. Explain: Possible answer: every time you make the first addend bigger, you have to make the second one smaller. you move over and down the same number •Focus students’ attention on the diagonal pattern of fives on the addition table. Remind students that the numbers along the top and sides of the table are the addends and that the sums are found inside the table. Be sure students understand how to use the table by having them put one finger on the addend 3 on the left and the addend 2 at the top and sliding their fingers together to see the sum of 5. of spaces and this makes a diagonal. 14 examine: Jace counted to 50 by fives. Annabel counted to 50 by tens. What numbers did both Jace and Annabel say? 10, 20, 30, 40, 50 Explain why some of the numbers they said were the same. Possible answer: some of the numbers are the same because there are 2 fives in a ten, so every other number will be said by both kids. 15 Determine: Pat saw an odd number of birds on Monday and an even number of birds on Tuesday. Is the total number of birds he saw odd or even? odd Explain how you know this, even though you don’t know how many birds he saw. Possible answer. an even number is always made of pairs of two. an odd •Have students work in pairs to complete the AddendAddend-Sum chart. •Ask students to explain why the 5s form a diagonal line. Expect students to say that the 5s are the sum and as one addend on the left gets bigger, the other added at the top gets smaller to make the sum of 5. number is always made of pairs of two with one left over. When you add an even and an odd, you have pairs of two with one left over. it doesn’t matter which even number and which odd number you have. 56 L7: Understand Patterns ©Curriculum Associates, LLC Copying is not permitted. Examine: •Ask two volunteers to come to the board and write multiples of five and ten to 50. Have one of the students circle the numbers they have in common. Ask students how patterns and some numbers are the same or similar in the two lists. Students should see that it takes two fives to make ten, so they will have some of the name numbers. Determine: •Students should recognize and understand that when two even numbers are added, the sum is even and when an even number and an odd number are added, the sum is odd. You may wish to discuss why when two odd numbers are added, the sum is even. 64 L7: Understand Patterns ©Curriculum Associates, LLC Copying is not permitted. Part 4: Common Core Performance Task Lesson 7 At a Glance Part 4: common core Performance task Students find patterns in a multiplication table and use them to complete the table. They describe a pattern and explain why the pattern works. Lesson 7 Put it together 16 Look at the multiplication table below. Step By Step •Direct students to complete the Put It Together task on their own. •Let students know that they can circle a pattern on the chart, but they need to describe the pattern in words for letter B. •As students work on their own, walk around to assess their progress and understanding, to answer their questions, and to give additional support, if needed. 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 2 0 2 4 6 8 3 0 9 12 4 0 4 8 5 0 5 10 15 20 25 30 35 40 45 6 0 6 12 18 24 30 7 0 7 14 21 28 8 0 9 0 3 8 9 6 12 16 10 12 14 16 18 15 18 20 24 24 27 28 32 21 36 36 42 48 54 35 42 49 56 63 16 24 32 40 48 56 64 72 18 27 36 45 54 63 72 81 a Fill in the missing numbers. b Describe a pattern you see in the table. Possible answer: all of the numbers in the 2 row are even. •If time permits, have students share and describe patterns they found. c Explain why the pattern works the way it does. Possible answer: all of the numbers are even because the row shows Scoring Rubrics groups of 2. 2 is even, and when you add even numbers, the sum is always even. A Points Expectations 2 B The student correctly fills in all the missing numbers. 1 The student correctly fills in some of the missing numbers. 0 The student does not fill in any missing numbers or incorrectly fills in all the missing numbers. Points Expectations 2 The student finds a pattern and clearly describes the characteristics of the pattern. 1 The student finds a pattern, but characteristics of the pattern are not fully described. 0 No pattern is described or what is described is not a pattern and the description is unclear. L7: Understand Patterns ©Curriculum Associates, LLC Copying is not permitted. L7: Understand Patterns ©Curriculum Associates, LLC C Copying is not permitted. 57 Points Expectations 2 The student shows understanding of the pattern and explains clearly why it works the way it does. 1 The student partially explains why the pattern works the way it does. 0 The student is unable to explain why the pattern works. 65 Differentiated Instruction Lesson 7 Intervention Activity On-Level Activity Find patterns in a multiplication chart. Further explore odd and even numbers using addition tables. Materials: blank multiplication chart Asking students to construct a multiplication chart can help them focus on and use patterns to complete the chart and also help them learn multiplication facts in the process. Ask students to fill in the first row of the table. Remind them that the first column also contains these same numbers and ask them to fill in the column. Point to the 2nd row and remind students that the numbers in this row are the multiples or “count bys” of 2. Ask them to fill in this row. Remind them that the column of twos contains these same numbers and instruct them to fill in the column. Have students fill in the fives and tens in the same way. Then move on to the threes, then the fours. Pause and direct students to look at the column and row of 5s and ask what they see repeating. [0 and 5] Ask whether each number is even or odd for the 5s. Have them circle the repeating zeros. Have them underline the repeating fives. Ask students to look for other patterns they see. Instruct students to circle or highlight several more patterns in the chart. Materials: A blank chart with 11 rows and 11 columns or a completed addition chart with addends to 10. Have students highlight the addends along the side and top of the table to make the sums easier to see. Instruct students to work in pairs or a group and use the addition table to help them explore questions, such as: If you add 2 even addends, will the sum be even or odd? If you add two odd numbers that are the same, will the sum be even or odd? If you add any two odd numbers on the chart, will the sum be even or odd? You may wish to give each group or pair a different question to explore. When pairs or groups share, ask questions such as: Can you give some examples? Do you agree with what said? Do you think this is true for all whole numbers? How do you know? Challenge Activity Materials: 1 inch (or larger) graph paper, scissors, tape, or use 2 colors of connecting cubes to make the train. Ask students to work in pairs or small groups. Instruct each student to color in this pattern for every 4 squares in one row on their graph paper: white square, white square, white square, colored square. Have the students cut the rows apart and tape them together to make one long train. Be sure that the entire 4-square pattern repeats throughout the train. The train should be at least 5 feet long. Ask students to put a Star on the first square to show that it’s the front of the train. Then instruct students to cut apart their train into squares of three starting the cutting from the front of the train. Have them cut at least 5 sections of 3 and stack them so the piece with the star is at the bottom of the stack. Have students notice and describe the pattern that the colored squares make (diagonal going from left to top right). Instruct students to put a star on the front of the remaining piece of train (the front square should be shaded) and have students cut apart about 4 or 5 pieces of train that are 4 squares long. Ask them to stack the pieces with the front of the train at the top. Ask students to predict what the stacked squares will look like if they cut the remaining train into sections of 5 squares. Then have students put a star on the front square and cut apart the remaining train into rows of 5, starting at the front of the train. Students can try this with cutting sections of 7 squares, and 8 squares. Have student notice whether any of the patterns repeat in the stacked squares. Ask if they can use any of the patterns to predict what trains of 10 and 11 would look like. 66 L7: Understand Patterns ©Curriculum Associates, LLC Copying is not permitted.