Features of the Teacher s Guide

Features of the Teacher s Guide Features of the Program Multiple units for the same domain. We return to topics as we need them for new material. By repeatedly revisiting the same concept throughout the year, we ensure that students have sufficient practice with each topic. This approach also emphasizes the connections between topics to deepen students understanding of each individual topic, as well as the coherent whole. For example, in Grade 4, the concept of times as many is taught first near the start of the year, then emphasized when students learn place value (each place value is worth ten times as many as the next smaller place value), practiced again when students learn multiplication (use 9 7 is three times as many as 3 7 to compute the former from the latter), and returned to when learning word problems. Lesson structure. Our lesson plans generally follow this structure: short explanation (at most 3 minutes) exercises and bonus questions incremental increase in challenge and another short explanation continue the cycle JUMP Lesson Short Explanation (up to 3 minutes) Practice Give a bonus Assess Increase level incrementally Scaffold The idea is that students who finish regular questions quickly can work independently on the bonus questions while struggling students work on the regular questions. Bonus questions are designed to require very little (if any) extra teaching. Not all students will need to do all the regular questions before attempting the bonus questions. When you see that a student is succeeding with questions at the level at which you want everyone to be, you can begin assigning the student bonus questions. Not all students will finish all the regular questions; even among students who are not ready to attempt bonus questions, some will be faster than others, and some will need more practice than others. Assign questions based on your students needs. The number of questions completed will vary with each student. Support for students with motor difficulties. In many lessons, students are asked to copy questions, to draw, or to cut or glue paper. Some students may need help or to have the copying, drawing, cutting, or gluing done for them. A-19

Grid paper. Use grid paper notebooks instead of regular lined notebooks for students. Grid paper is useful for lining up digits, making tables, drawing shapes such as rectangles, coordinate planes, and so on. It is also an invaluable tool for students with diagnosed and undiagnosed problems in visual organization. Using it for all students will help these students without singling them out. One-quarter-inch grid paper is fine for most uses. Blackline Masters are provided when larger grids are required. Strategies for immediate assessment (signaling). When a problem has a simple answer, such as one word, a short phrase, or a sign (e.g., + or ), students can be asked to signal the answer. For example, if the answer to a question is yes or no, students can signal thumbs up for yes and thumbs down for no. Or, if a student has given an answer, the others can signal agreement with a thumbs up or disagreement with a thumbs down. Students can also signal answers by making shapes or signs. For example, if the potential answer to a question is + or, students can make the sign they choose with their fingers or hands. Signaling is also useful when you have multiple choice questions: number the answers and have students hold up the number of fingers corresponding to the answer they think is correct. You need to insist that students signal their answers at the same time, to minimize copying other answers. One way to do so is to give students adequate thinking time and then have students all show their answer on the count of three. Make sure students are familiar with the structure before using it in content-learning situations. Have students practice signaling the answers at the same time beforehand. Group work. Be sure to structure group work to encourage collaboration, fairness, and effective use of time. When students fulfil different roles in each group, all roles need to be meaningful, and it is best to rotate the roles among the participants in a group. Another way to ensure effective use of time and fairness is to give each student the same amount of time to speak by timing them (many countdown timers are available online) and telling them when to switch speakers. Encourage students to listen by asking them to summarize what other people have said before they have their turn, or to provide meaningful encouragement, praise, and feedback to the previous speaker. However, you need to explicitly teach, demonstrate, and practice group work skills, such as using a timer, summarizing, or giving praise and feedback, before they can be used in content-learning situations. Games. Like the other activities in the lesson plans, the games provided require individual work (working alone) or cooperation (working with a partner) instead of competition (working against an opponent). Points are sometimes awarded in a game, but only toward a team score. It is important not to ask individual teams what their point total is, since this question would promote competition among teams. There are several benefits to individual and cooperative activities. Benefits of Individual Activities 1. Maximizing engagement. When students are working individually, there is no off-time as there is when they are waiting for a partner to do a task. 2. Building confidence for group work. Working individually prepares students to work in groups. Only once students can successfully complete individual tasks will they have the confidence to share their work in group settings. 3. Facilitating assessment. It is easier to assess each student when they are doing individual work because you know exactly whom you are assessing. A-20

Benefits of Cooperative Activities 1. Building teamwork skills. Cooperating with others is an essential skill outside the classroom and throughout life. 2. Accelerating learning. In the setting of a cooperative game, players align their goals in order to win. When students are allowed to use their energy to pull each other along in the same direction, rather than work against each other, students are more likely to progress faster. 3. Promoting empathy. Empathy (understanding the feelings of others and compassionately acting on that understanding) has been shown effective in reducing bullying. When working towards common goals, students naturally mirror each other s emotions, an important aspect of empathy. 4. Avoiding hierarchy. When students succeed or fail together, it is less likely that some students will be considered superior to others. Avoiding hierarchy promotes the collective excitement that sweeps through a class when everyone experiences success together. Problems and Puzzles. At the end of certain units, we provide a Problems and Puzzles page. The page usually does not require a lesson, but is intended as a review and to ensure that students can combine the concepts learned and can use them in new ways. Using the AP Book as a diagnostic tool. Some lessons are labeled as review. In these lessons, you could have students do the pages as a diagnostic assessment and then teach the material as necessary. Be sure, however, that students work at the same pace. There is no point in having some students finish the exercises in five minutes while others need the whole class period. Use the bonus questions in the teacher s guide to keep students who finish early busy. Break the student pages up into chunks and do not allow any student to work ahead until all students have finished each chunk. Features of Lesson Plans Goals. The specific goals for student learning are stated at the beginning of each lesson plan. Prior knowledge required. Any prior knowledge that students need in order to understand the concepts taught in a lesson is reviewed at the beginning of the lesson. The prior knowledge required is also listed at the beginning of each lesson plan. If you decide to teach the lessons in a different order than that presented in the AP Book, it is essential that you pay close attention to this list, to ensure that you have covered all the necessary background material. Materials. The materials required for each lesson are listed at the beginning of the lesson plan. In grades 4 and 6, this list includes any materials you will need for Activities, but not Extensions; any materials needed for an Extension are listed within the Extension. In grade 5, materials for all parts of the lesson are listed at the beginning. Vocabulary. We list vocabulary words at the beginning of each lesson plan. Make sure students are familiar with the vocabulary words. Make some of the words, such as geometrical terms, part of your spelling lessons. Standards. The Common Core State Standards covered by the lesson are stated at the beginning of each lesson plan. A-21

Mathematical practices. We guide students to develop the Mathematical Practice Standards, by explicitly teaching the skills required. While the development of these practices occurs in virtually every lesson, only some lessons have grade level applications of the standards. These grade-level applications have been identified in the margin. Descriptive subheadings. Subheadings in boldface summarize the main parts of the lesson and the order in which teachers should introduce concepts and work through different types of problems. Support materials Blackline masters (BLMs) provide: materials needed for games (e.g., game boards and playing cards). replacements for resources that you may not have in your classroom (e.g. hundreds charts, 1-cm grid paper, nets of 3-D shapes). extra questions for remedial practice, grade-level practice, or advanced practice. Quizzes and tests SMART Board-compatible interactive whiteboard lessons A-22

Sample AP Book Pages NOTE: These are composite Grade 4 pages. They differ slightly from the actual pages. Grade Lesson number Lesson title Domain OA = Operations and Algebraic Thinking NBT = Number and Operations in Base Ten NF = Number and Operations Fractions MD = Measurement and Data G = Geometry Reminder boxes contain summaries of information taught previously. MD4-11 Length (Review) REMINDER 1 cm = 10 mm 1 m = 100 cm 1 km = 1,000 m 1. Fill in the tables. a) m cm b) cm mm c) km 1 2 3 25 1 2 3 37 1 2 3 75 m 2. Mark the measurements on the number line. (First convert all measurements to cm.) A. 150 cm B. 2 m C. 1 m 0 cm 50 cm 100 cm 150 cm 200 cm Illustrations are used primarily to help students visualize the problems. 3. This table shows the lengths of some animals at the zoo. a) Mark the lengths of L, R, B, and W on the number line. 0 cm 100 cm 200 cm b) How many centimeters longer than a lynx is a wolf? 4. A fence is made of four parts joined end to end. Each part is 32 cm long. 32 cm Animal Lynx: L Rabbit: R Beaver: B Wolf: W Length 150 cm 50 cm 100 cm 2 m Most questions require short answers to minimize copying and facilitate easy checking. (The lesson plans contain more complex questions). Question numbers are in bold so that they are easy to find. 158 a) How long is the fence in centimeters? b) Is the fence longer or shorter than a meter? 5. The distance from Joan s home to school is 600 m. The distance from school to the library is 700 m. Joan walked to school and continued to the library. a) How many meters did she walk? b) Is her walk longer than a kilometer or shorter than a kilometer? c) How many meters longer or shorter than a kilometer is Joan s walk? Measurement and Data 4-11 Domain, grade, and lesson number. A-23

To multiply 20 60, Solmaz multiplies 2 (10 60). The picture shows why this works. 60 Partial or complete answers (including intermediate steps where applicable) appear in italics. 10 20 10 Two 10 60 rectangles make one 20 60 rectangle. 9. Multiply. a) 20 60 = 2 (10 60) b) 30 500 = 3 (10 500) c) 40 800 = 4 (10 800) = 2 600 = 3 = 4 = 1,200 = = d) 30 80 = 3 (10 80) e) 70 600 = 7 (10 600) f) 60 700 = 6 (10 700) = 3 = 7 = 6 = = = Teaching boxes contain definitions, explanations, examples, and step-by-step instructions. To multiply 40 700: Step 1: Multiply 4 7 = 28. Step 2: Write all the zeros from 40 and 700 in the answer: 40 700 = 28,000. 10. Multiply the 1-digit numbers to multiply the tens and hundreds. a) 8 5 = 40 b) 2 3 = c) 5 2 = 800 50 = 40,000 20 300 = 50 200 = Bonus questions for students to work on independently, without additional teaching. The notebook icon indicates questions that students must answer in a notebook. 104 d) 8 7 = e) 4 9 = f) 5 6 = 800 70 = 40 9,000 = 500 600 = g) 40 30 = h) 300 50 = i) 80 500 = j) 800 900 = k) 50 5,000 = l) 40 50,000 = BONUS 3,000 80,000 = 11. a) Calculate each product on a calculator. i) 3,142 608 ii) 2,984 497 iii) 70,162 811 = = = b) Use estimation by rounding to check if your answers in part a) make sense. Explain how you know. Use words such as close to, higher than, and lower than. Number and Operations in Base Ten 4-36 A-24

Sample Lesson Plan Pages NOTE: These are sample Grade 4 lesson plan pages. They differ from the actual lesson plans. Grade Lesson number Domain OA = Operations and Algebraic Thinking NBT = Number and Operations in Base Ten NF = Number and Operations Fractions MD = Measurement and Data G = Geometry Lesson title Pages in the AP Book related to this lesson. NBT4-49 Multiplying 2 Digits by 2 Digits Page 123-125 Common Core State Standards (may include standards from prior grades, where appli cable), stated explicitly. Goals STANDARDS 4.NBT.5 The purpose of the lesson, stated explicitly. Students will multiply 2-digit numbers by 2-digit numbers. PRIOR KNOWLEDGE REQUIRED VOCABULARY Can multiply a 2-digit by a 1-digit number Can multiply a 2-digit number by a 2-digit multiple of 10 double multiple Test and activate prior knowledge before using it. MATERIALS grid paper Introduce the lesson topic. Write on the board 28 36. ASK: How is this multiplication different from any we have done so far? (we have never multiplied a 2-digit number by a 2-digit number of which neither is a multiple of 10 we have only estimated the product in such cases.) Explain the meaning of these terms and write them on the board as they appear in the lesson. Include them in your spelling tests from time to time. (MP.3) Splitting a problem into easier problems. Tell students that you would like to think of a way to split the problem into two easier problems, both of which they already know how to do. Have students list all the types of problems they know how to do that might be helpful: Multiply a 1-digit number by a 1-digit number. Multiply a 2-digit number by a 1-digit number. Multiply a 2-digit number by a 2-digit multiple of 10. Allow students time to think of a way to split the problem into two easier products that they already know how to do. Possibilities include: 28 36 = (20 36) + (8 36) 28 OR 28 36 = (28 30) + (28 6) Read these out loud as 20 thirty-sixes plus 8 thirty-sixes or 30 twentyeights plus 6 twenty-eights. 6 30 Write the equations out with blanks if students need a prompt. Example: Manipulatives and teaching aids that need to be prepared ahead of time are listed. 28 36 = 28 + 28 SAY: 28 is a 2-digit number. How can we split 36 into 2 numbers so that we know how to do both products? Draw the picture in the margin on the board. (MP.2) Remind students that the area of the whole rectangle is the sum of the two smaller rectangles. Write on the board: 28 36 = (28 30) + (28 6), and SAY: 36 twenty-eights is 30 twenty-eights plus 6 twenty-eights. Also remind students that we write brackets to show what operations we do first. SAY: We first find the areas of the two smaller rectangles (point to the two products as you say this), and then we add them together to get the area of the whole rectangle (point to the addition as you say this). Teach the concept explicitly, using brief explanations (up to 3 minutes). Practice adding the two parts. Write on the board: 23 7 = 161 and 23 50 = 1,150. ASK: What is 23 57? (161 + 1150 = 1,311) Number and Operations in Base Ten 4-49 E-1 A-25

Exercises for individual practice (with answers in some cases) are highlighted. Bonus questions that students can solve independently allow you to spend more time with struggling students. Specific prompts which encourage thinking about or communicating the idea are called out. 20 7 30 9 40 3 40 7 (MP.7, MP.2) 50 2 3 20 3 7 Now write the 23 times table on the board: 23 23 23 23 23 23 23 23 23 1 2 3 4 5 6 7 8 9 23 46 69 92 115 138 161 184 207 ASK: What is 23 6? (138) What is 23 60? (1,380) How do you know? (because 23 60 is 23 6 10) What is 23 67? (1,380 + 161 = 1,541) Have students use these facts to multiply 23 by various 2-digit numbers. Exercises: 23 45; 23 78; 23 46; 23 64. Answers: 1,035; 1,794; 1,058;1,472. If students need scaffolding, provide the following structure, either on the board or to individual students (to find 23 45): 23 4 = so 23 40 = and 23 5 = so 23 45 = Bonus: 23 4 = so 23 400 = and 23 5 = so 23 405 = Now draw a 43 27 rectangle on the board and have students copy the rectangle and separate it into two smaller rectangles so that the area of each is easier to find. Take different answers. (43 20 and 43 7 OR 40 27 and 3 27) Have students find the areas of the smaller rectangles and then add them together to find the area of the larger rectangle. Have students use this method to find several products of 2-digit numbers. Exercises: 54 45; 36 44; 47 68, Bonus: 43 502. Answers: 2,430; 1,584; 3,196, Bonus: 21,586. Using area to divide a product of 2-digit numbers into four easy products. Remind students that even 43 20 was a combination of smaller products: 40 20 and 3 20. ASK: How can we write 43 7 as a combination of smaller products? (40 7 and 3 7) Summarize by saying that 43 27 is actually a sum of four very easy products: 43 27 = (40 20) + (3 20) + (40 7) + (3 7) = 800 + 60 + 280 + 21 = 1,161 Draw the first picture in the margin on the board to summarize. Then draw the second picture. ASK: What product can you determine by finding the area of the big rectangle? (52 39) PROMPT: How long are the sides of the rectangle? What products can you determine by finding the areas of the small rectangles? Give students time to write down the four products. Have students check their answers with a partner. Then take up the answers: 50 30; 2 30; 50 9; and 2 9. ASK: What makes these four products so easy to find? (we just have to know how to multiply single-digit numbers, and multiples of 10) Have students use the four smaller products to find 52 39 then check their answers with a partner. Point out before they check with a partner that if they do not get the same answer, they should then check to see if one of the four products is different, and then if they are all the same, look at the addition. E-2 Teacher s Guide for AP Book 4.1 Provide tasks for practice and individual or at-a-glance assessment. Make the step smaller for students who need help, or explain in a different way. Provide bonus questions to raise excitement and to keep faster students engaged. Foster independence by providing strategies to find mistakes. A-26

Opportunities to develop Mathematical Practices are flagged. (MP.8) Relating the standard algorithm to using the sums of the four easy products. Now show students the solution to this problem using the standard algorithm: 1 52 becomes 52 39 39 18 (2 9) 468 (52 9) 450 (50 9) + 1,560 (52 30) 60 (2 30) 2,028 + 1,500 (50 30) 2,028 The main idea or concept behind each part of the lesson is in bold at the beginning of a paragraph. Sample answers are often provided. (MP.3) ASK: Where does the 1 (point to the 1 written above the 5) come from? (2 9 is 18 which is 1 ten and 8 ones, so we write 1 in the tens column and 8 in the ones column.) How do we use the 1 when multiplying 52 39? (When we multiply 50 9 = 450 = 45 tens, we add 1 to the number of tens, so now we have 46 tens) Exercise: Have students show how each of the four products that are added were obtained: 73 49 27 ( ) 630 ( ) 120 ( ) + 2,800 ( ) 3,577 Then have students rewrite the problem using the standard algorithm notation. Ensure that students write the regrouping in the correct column. Notice that students will need to show regrouping twice: once when multiplying 3 9 (carry the 2 to the tens column) and once when multiplying 3 40 (carry the 1 to the hundreds column). This will be easier for students to do if they use grid paper. Repeat for other products, first having students write how the four products were obtained, then having them write the product using the standard algorithm notation. Exercises: 46 58; 34 29; 67 76. Answers: 2,668; 986; 5,092. Exercises: Use the standard algorithm only to do these problems: a) 73 46 b) 54 35 c) 46 71 d) 84 96 Answers: a) 3,358, b) 1,890, c) 3,266, d) 8,064 Bonus Add enough zeros to a product of two 2-digit numbers to multiply: a) 780 640 b) 3,400 250 c) 8,700 9,400 Answers: a) 499,200, b) 850,000, c) 81,780,000 12 73 49 657 + 2,920 3,577 Number and Operations in Base Ten 4-49 E-3 Raise the bar incrementally. Repeat the cycle: assign a task, assess, provide scaffolding and/ or bonus questions. A-27

Hands-on or whole-class activities help students consolidate their knowledge. Estimating sums and products. Explain to students that they don t always need an exact answer, but often only need to know about how big the answer is. Do some examples together. For example, 32 + 85 is about 30 + 90 = 120, and 32 85 is about 30 90 = 2,700 (using rounding). ACTIVITY Hot and cold. This is a game for pairs. Player 1 picks two numbers between 50 and 100 and estimates the product of these numbers. Player 2 uses a calculator to find the actual product and gives Player 1 an appropriate clue about the estimate, using the words hot, warm, warmer, cold, colder, freezing, and so on. Player 1 revises his or her estimate until it is burning hot, or within 100 of the correct answer. Players switch roles and play again. Structure group work using time limits, meaningful roles, and rotating tasks. Practice skills engagingly. Ideas for at-a-glance assessment are provided. (MP.6) Checking the reasonableness of an answer when using a calculator. SAY: John multiplied 32 86 on a calculator and got 1,978. How can he tell he is wrong? (the answer should be more than 30 80 = 2,400, so the answer is too low) Explain to students that John input 23 86 by mistake. This is a type of mistake anyone can make, even if you know the math, so it is really important to check that your answer makes sense. Now tell students that some of the following products were input incorrectly into a calculator. See if they can tell which ones by estimating. Suggest that students round both numbers up to get a high estimate and both numbers down to get a low estimate. Then they can be sure that their answer should lie in between the two estimates. a) 27 52 = 3,744 b) 38 94 = 1,862 c) 12 74 = 888 Answers: a) and b) were input into the calculator incorrectly. Students can signal their answers showing thumbs up when the product is reasonable, and thumbs down when it is not. Encourage students to check their estimates against the actual products (found using a calculator). Have students use technology when they understand its value, and teach them to monitor results obtained by technology for possible mistakes. Extensions Optional extensions provide extra challenges. Addition Multiplication 32 + 85 32 85 41 + 24 41 24 36 + 54 36 54 72 + 41 72 41 72 + 49 72 49 89 + 51 89 51 39 + 68 39 68 45 + 56 45 56 32 + 87 32 87 33 + 82 33 82 1. a) Give students 20 estimation problems to do, including 10 additions and 10 multiplications (2-digit by 2-digit). See example in margin. Students could check their estimates using an online tool. See http://teacherlink.org/estimate for one such tool. Students could record the number of good estimates they produced for the 10 addition problems and the 10 multiplication problems on their first attempts, or they could count how many attempts they needed to estimate correctly. Either way, students will get an idea of whether they are better at estimating sums or products. b) Have students construct a test to determine if they are better at estimating products of two 2-digit numbers or products of a 3-digit number and a 1-digit number. E-4 Teacher s Guide for AP Book 4.1 A-28

Some lessons might require the use of Blackline Masters. Page references are provided. (MP.8) 2. On BLM Patterns in Multiplication (p. E-80), students discover an easy way to multiply a 2-digit number having ones digit 5 by itself: 15 15, 25 25, 35 35, and so on. After students complete the BLM, summarize their answers. ASK: What are the tens and ones digits always? (25) How can you get the number of hundreds in the answer from the tens digit of the number being multiplied by itself? (multiply the tens digit by 1 more than the tens digit) Explain to students that they can now multiply some 2-digit numbers in their heads (and this is a shortcut that not even most mathematicians know about!) Tell students to not look at the answers they just wrote down. Write on the board: 35 35 = ASK: What are the last two digits? (25) How do you know? (because they are always 25 that is easy to remember) What are the first two digits? (3 4 = 12) How do you know? (because that is the pattern we found; 4 is one more than 3, so multiply 3 4) Show this on the board as follows: 35 35 = 1,2 2 5 3 4 Have students do these questions mentally: a) 75 75 = b) 65 65 = c) 45 45 = d) 85 85 = e) 95 95 = If students are engaged, you could tell them that this same shortcut works for multiplying any number with ones digit 5 by itself. Show this on the board: 175 175 = 30,625 Domain, grade, and lesson number. Number and Operations in Base Ten 4-49 17 18 Challenge students to calculate these products: a) 105 105 b) 995 995 c) 1,005 1,005 d) 9,995 9,995 Answers: a) 10 11 = 110, so 105 105 = 11,025, b) 99 100 = 9,900, so 995 995 = 990,025 c) 100 101 = 10,100, so 1,005 1,005 = 1,010,025 d) 999 1,000 = 999,000, so 9,995 9,995 = 99,900,025 Encourage students to check the reasonableness of their answers by estimating using rounding. For example, a) should be a little more than 100 100 = 10,000, which it is. E-5 A-29