NS2-45 Skip Counting Pages 1-8

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1 NS2-45 Skip Counting Pages 1-8 Goals Students will skip count by 2s, 5s, or 10s from 0 to 100, and back from 100 to 0. Students will skip count by 5s starting at multiples of 5, and by 2s or 10s starting at any number. PRIOR KNOWLEDGE REQUIRED can count by 1s can use a hundreds chart to count can use a hundreds chart to add or subtract 10 can use a number line can measure centimetres with a ruler VOCABULARY skip skip counting count back how many hundreds chart number line more/ less ones digit tens digit centimetres MATERIALS flip chart rulers metre sticks large hundreds chart in school yard cards numbered 1 through to the number of students counters BLM A Larger Hundreds Chart (p xxx) CURRICULUM EXPECTATIONS Ontario: 1m21, 2m4, 2m6, 2m7, 2m19, 2m20, WNCP: 1N1, 1N3, 2N1, [CN, C, V, R] Introduce skip counting by skipping every second number. SAY: I want to count every second number. This is called skip counting by 2s. Write the words skip counting on the board. To demonstrate, SAY: 0 (loudly), skip 1 (quietly), 2 (loudly), skip 3 (quietly), 4 (loudly), and so on to 10. Repeat with students. Explain that when you skip count by 2 you add 2 instead of 1 to find the next number. Skip count using a chart. Draw the first row of a hundreds chart on a flip chart. Have the class slowly skip count by 2s. Colour the numbers as the class says them:

2 Cover up the chart and ask students if 3 is a number they say when counting by 2. (thumbs up for yes, thumbs down for no) Then uncover the chart to check. Repeat for other random numbers. Eventually stop checking answers and increase the speed at which you say the numbers. Count by 2 up to 100. Draw the first two rows of a hundreds chart on the board and ask a volunteer to show skip counting by 2 by shading every second square. Ask students if they see anything the same about the numbers in the first row and the numbers in the second row. PSS Looking for a pattern. ASK: If I can skip count by 2 up to 10, does that help me skip count up to 20? How? (The numbers up to 20 have the same ones digit as the numbers up to 10.) Does it help me skip count to 100? (yes, the ones digits are again the same as skip counting up to 10) Activity 1. Skip count on a large hundreds chart in the school yard. Have students hop on every second number, starting at 2 and chant the numbers as they land on them. When the first student says 12, the next student says 2. Count back by 2s from 100 to 0. Teach students to skip count back by 2s from 10 to 0 by memorizing the sequence. Then teach them to skip count back by 2s from 20 to 10 by using the same pattern in the ones digits. Repeat for 30 to 20, and so on. Repeat Activity 1 but start at 100 and go backwards. Count by 2s from 1. As a class, count by 2s starting at 1. SAY: 1 (loudly), skip 2 (quietly), 3 (loudly), and so on. Then have students count by 2s starting at 1 on the first 2 rows of a hundreds chart. Activity 2. Catch. (See NS Part 1 Introduction) Throw a number from 0 to 100, (include both odd and even numbers), to one student after another. The student says the next number counting by 2s (i.e. the number that is two more than). Count back by 2s starting from any number. Repeat Activity 1 but start at 99 and go backwards. Then teach this the same way you taught counting back by 2s from 100. Finally, repeat Activity 2, but this time the student says the next number counting back by 2s (i.e. the number that is two less than). Count by 10s from 0 to 100 using a hundreds chart. Remind students that to add 10, they simply move down a row. Emphasize that counting by 10s is the same as counting by 1s, except that they just add a 0 to the numbers they say when counting by It may be helpful for some students to notice the similarities in the sound of the numbers as well: e.g., three and thirty, four and forty. Repeat the previous activities counting by 10s instead of by 2s. Then have students check their answers for what comes next by providing them with the number that comes after that. PSS Reflecting on the reasonableness of an answer

3 EXAMPLES: 30 50; 10 30; 70 90; Students should guess that 40 comes after 30 and then check that 50 comes after 40 when counting by 10s. Provide students with BLM A Larger Hundreds Chart and have them count by 10s from various numbers (all with ones digit 0). Remind students that they can move down a row to add 10. EXAMPLE: Start at 30, and count 30, 40, 50, 60, 70, 80, 90, 100. Count back by 10s from 100 to 0. Using a hundreds chart, move up a row instead of down a row. Or count back by 1s instead of forward by 1s to help. Count by 10s starting from any number. When counting forwards or backwards, the last digit stays the same. The number of tens goes up or down by 1. Start at any number and go up or down a row in a hundreds chart. Use a metre stick to count forward or backward by 10s. Use a ruler at least 10 cm long, preferably exactly 10 cm long. EXAMPLE: To count back by 10s from 83, place the 10 cm mark on the 83. Where is the 0 mark? (on 73) Continue counting back in this way. (63, 53, 43, 33, 23, 13, 3) Compare using a hundreds chart and a metre stick. ASK: Which way makes counting back by 10s easier? Why? PSS Selecting tools and strategies Count by 5s from 0 to 100. To count by 5s, repeat the lesson for counting by 10s. Emphasize that counting by 5s is easy once they know how to count by 10s. SAY: After you count 0, 5, 10, 15, say the same numbers you would say counting by 10s, and then repeat that number once more with a five after it: 20, 25, 30, 35, and so on. Activity 3. PSS Organizing data, Visualization Give each student one number card from 1 to the number of students. Have students order themselves starting at the student with card number 1. Then tell students to shuffle themselves. When students are well-shuffled, have them order themselves by first placing those who have cards that count by 5s (0, 5, 10, 15, 20); the remaining students can then place themselves in-between where they belong. Discuss which way was easier, which way took longer, and why. Count back by 5s from 20 to 0. Teach students to memorize the sequence (20, 15, 10, 5, 0). As a class, say the sequence forwards to 10 (0, 5, 10) then backwards (10, 5, 0), then forwards to 15 (0, 5, 10, 15), then backwards (15, 10, 5, 0), then forwards to 20 (0, 5, 10, 15, 20), then backwards (20, 15, 10, 5, 0). Count back by 5s from 100 to 0. Emphasize that counting back by 5s is easy once they know how to count back by 10s; say the same numbers you would say counting back by 10s, but before saying a number, say it first with a five after it. For example, after saying 30, 20 would be next counting by 10s, so say 25 first, then 20. Show this on a number line or hundreds chart. Grouping objects to count them. Show students a large pile of counters, say 57 counters, and tell them that you want to count how many there are. Start by counting 1, 2, 3, 4, and so on. Demonstrate making a mistake partway through and explain that you

4 need to start over because you forgot where you are in counting. Then count 5 at a time and put them in groups of 5. Invite volunteers to help. Explain that you find it easier to count to 5 because at least you won t get lost in the counting. Once you ve grouped the pennies into groups of 5, with 2 leftover, explain that now you can use skip counting to find how many there are. Since there are 5 in each group, you can skip count by 5s. Do this together as a class. 5, 10, 15, 20, and so on, until 55. SAY: There are 55 here and 2 more. How many is that? Now we have to count by 1s because we no longer have groups of 5. Write the counting sequence on the board: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 56, 57. Extensions: 1. Find the mistake in skip counting. Ontario teachers might use the optional lesson PA2-14: Finding Mistakes as an extension here. Do not use it as an extension here, though, if you plan to teach it later. 2. Combine skip counting by 2s and skip counting by 10s to skip count by 20s. To count by 20s, count by 10, but skip every second 10. (SAY: skip 10 (quietly), say 20 (loudly), skip 30 (quietly), say 40 (loudly), and so on.) Bonus: Have students skip count from 400 to 500 and then from 560 to On BLM Skip Counting (p xxx), students will discover that the numbers they say when counting by 10s are the numbers they say both when counting by 2s and by 5s. 4. Using skip counting to add many numbers. Have students pair up the numbers that add to 5 or 10 to add many numbers. EXAMPLES: = = 20 (skip count 5, 10, 15, 20) = = 70 (skip count 10, 20, 30, 40, 50, 60, 70) Provide BLM Adding Many Numbers (p xxx). 5. PSS Guessing, checking and revising Teach students what to count by given the first and last numbers, and the number of spaces in between. Draw a number line like this: SAY: I want to skip count from 10 to 14 and I want to say only one number in-between. If I count by 1s, I know that 11 is right after 10, but 14 doesn t come right after 11, so that won t work. ASK: What should I skip count by if I want the same number to come right after 10 and right before 14? (count by 2s) Check this answer by skip counting from 10 to 14. Repeat with more examples, gradually leaving more spaces between numbers. Have students progress as follows: first choose between skip counting by 2s or 5s, then choose between skip counting by 5s or 10s; finally choose between skip counting by 2s, 5s or 10s. Students should guess what to skip count by and check their guess if, when skip counting by their guess, they don t get to the end number, students should decide whether to skip count by a higher or lower number based on the results.

5 AT HOME House numbers CONNECTION Literature, Two of Everything by L.T. Hong A Chinese folktale counts everything by 2s.

6 NS2-46 Closer To Pages 9-12 Goals Students will determine the closest ten by using the distance on a number line. PRIOR KNOWLEDGE REQUIRED understands the concept of distance can count to 100 knows how to use number lines VOCABULARY close/ closer/ closest more/ less between far apart further equally MATERIALS large cards numbered 0 through 10 a large visible hundreds chart (e.g. a pocket chart) BLM Closest (p xxx) BLM Number Lines 0 to 10 (p xxx) BLM Closer to 0 or 10 (p xxx) BLM Closer to 40 or 50 (p xxx) CURRICULUM EXPECTATIONS Ontario: 2m1, 2m6, 2m7, 2m14 WNCP: 2N6, [V, C, R] Review the words closer and further, closest and furthest. Choose two volunteers sitting clearly at different distances from the front of the classroom. Have them to stand up. ASK: Who is closer to the front? Who is closer to the back? Who is further from the front? Continue with distance from objects in the room. EXAMPLES: from the bookcase, the teacher s desk, filing cabinet. Then have four volunteers. Ask similar questions regarding one volunteer s position to another. EXAMPLES: Who is furthest from Bonnie? Who is closer to Teah than Julia is? EXTRA PRACTICE BLM Closest Determine which pair of dots is closer together. Draw the picture in the margin on the board. Have students decide which pair of dots above the line or below it is closer together. Remind students that closer means more close. Ask students to brainstorm other words where the er ending means more (e.g. long and longer, fast and faster, hot and hotter). Then continue having students decide which pair of dots is closer together for various examples. EXAMPLE:

7 Compare how close or far apart numbers are by looking at a number line. PSS Modelling Draw a number line from 0 to 6 on the board. ASK: Is 3 closer to 1 or to 2? Draw dots at 1 and 3 above the number line and at 2 and 3 below the number line (see the margin for an example). Example: Is 3 closer to 1 or to 2? Which number any number is closer to won t depend on the number line drawn. Draw two number lines from 0 to 6 that both compare how close 4 is to 1 with how close 4 is to 2, but change the spacing of the numbers on the number line: ASK: Is 4 closer to 1 or to 2 on this number line? (point to the first number line) How about on this one? (point to the second number line) Explain that no matter how we draw the number line, 4 is always closer to 2 than to 1. Mathematicians say that 4 is closer to 2 than to 1 because that s how it is on any number line. All numbers on a number line must be the same distance apart. Draw a number line where 0 to 7 are very close together and 7 to 10 are very far apart, so that 7 is closer to 0 than to ASK: Is 7 closer to 0 or to 10 on this number line? (to 0) Why did that happen? SAY: When mathematicians say that 7 is closer to 10 than to 0, they mean that 7 will be closer to 10 than to 0 on any number line, as long as all the numbers are the same distance apart. Draw two different correct number lines, with all the numbers the same distance apart, on the board to illustrate. On both of them, 7 is closer to 10 than to 0. Closer to means fewer numbers away from. Have eight volunteers stand in line at the front of the room, so that there is a clear front to the line-up. ASK: Is Jamie closer to the front of the line or to the back? How do you know? How many people are in front of him? Behind him? (Possible answer: More people are behind him than in front of him, so he is closer to the front.) Is Wei closer to Hamide or to Mina? (Possible answer: Wei is closer to Mina, because only one person is between him and Mina but 3 people are between him and Hamide.) Repeat with 11 volunteers each holding a large card from 0 to 10, to form a number line. Instead of naming students, refer to the students by the number they are holding. For example, ASK: Is the person holding 3 closer to the person holding 7 or to the person holding 1? Is 8 closer to 5 or to 10? Emphasize that two numbers are closer together if fewer numbers are between them.

8 EXTRA PRACTICE BLM Closer to 0 or 10 Is the number closer to 0 or to 10? Provide each student with a number line from 0 to 10 (see BLM Number Lines 0 to 10). Ask students to determine which numbers are closer to 10 than to 0 (6, 7, 8, 9), which numbers are closer to 0 than to 10 (1, 2, 3, 4), and which number is equally close to both. (5) ASK: Which numbers are more than 5? (6, 7, 8, 9) Which number are they closer to 0 or 10? Repeat for numbers less than 5. Numbers with ones digit more than 5 are closer to the next ten. PSS Looking for a pattern. Draw two number lines on the board, one from 0 to 10 and the other, right underneath it, from 30 to 40. Or, if you have a pocket chart with 11 columns, so that the tens appear on both the left and right sides, draw your students attention to it. ASK: Is 38 closer to 30 or 40? How do you know? (because 8 is closer to 10 than to 0, so 38 is closer to 40 than to 30) Point out that since the ones digit is more than 5, the number is closer to the higher 10. Repeat with more examples, and then have students do more individually: EXAMPLES: Is 43 closer to 40 or 50? Is 76 closer to 70 or 80? Determine the closest ten. Have students progress as follows. First, have students list the numbers that are between two given tens (e.g. between 30 and 40); then, have students find the tens that a given number is between. Volunteers may demonstrate some of their answers by finding the number on a large hundreds chart (for example, a pocket hundreds chart) and show the two tens it is between. EXAMPLE: What two tens is 73 between? (70 and 80) Finally, have students determine the closest ten. Once you know which two tens 73 is between, look at the ones digit, 3, to determine which ten it is closest to 3 is less than 5, so 73 is closer to 70 than to 80. EXAMPLES: 49, 77, 12, 84 (see margin for example). Students can check their answers using measuring tape or a metre stick. EXAMPLE: 9 is than 5 49 is between and 49 is closest to EXTRA PRACTICE BLM Closer to 40 or 50

9 NS2-47 Estimating Numbers Page xxx Goals Students will repeatedly guess and revise estimates based on grouping objects in tens. PRIOR KNOWLEDGE REQUIRED can group objects in bundles of 10 can count objects to 10 can count by 10s understands place value to tens and ones (e.g. 50 = 5 tens) VOCABULARY estimate check closest array about MATERIALS many straws cut in thirds many pennies many similar-sized beads a full jar of jelly beans an empty jar same size as jelly bean jar cards with random dots (see below) BLM Jelly Beans (p xxx) BLM Quantity 5 or 10 BLM Quantity 10 or 20 CURRICULUM EXPECTATIONS Ontario: 1m17, 2m1, 2m3, review WNCP: 1N6, 2N6, [ME, V] Estimating means guessing by using information. You will need the cards from BLM Quantity 5 or 10. Show students the back side of one of the cards and tell students to guess whether the number of dots on the other side of the card is closer to 5 or 10. Now turn the card around for a short time and have them guess again whether the number of dots is closer to 5 or 10. Finally, count the dots to decide together if the number of dots is closer to 5 or 10. Repeat with the remaining cards. ASK: Were your guesses more accurate when you saw the cards? SAY: When we guess based on information instead of just wild guessing, we are estimating. Write the word estimate on the board. Repeat with the cards from BLM Quantity 10 or 20, but this time tell students to decide whether the number of dots is closer to 10 or 20. Again have students guess blindly at first for each card, and then have them look quickly at the cards to estimate. Review grouping objects to count them. For example:

10 There are 10, 20, 30, 31, 32, 33, 34, 35, 36, 37 dots. Students will need to do this on Workbook p. 13. If students struggle with this, encourage them to write the tens above the groups of ten as they count them. Estimate how many by grouping 10. PSS Guessing, checking and revising EXAMPLE: Give each student a large handful of straws (cut in thirds) and have them guess how many straw pieces they have. Take a bundle of 40 straws yourself and model guessing 21. Have students bundle one group of 10 straws with elastics and guess again. Model doing so yourself; explain how you know that 21 is not reasonable anymore there is a lot more than 10 left after bundling 10. Repeatedly bundle ten straws and repeatedly revise the guesses. SAY: You used more and more information as you made more guesses. Even your first guess wasn t a completely wild guess because you were using some information. For example, no one guessed 3 straws because you could see that you had many more than 3. So, you were always estimating, but your guesses got better and better because each guess used more and more information. Repeat with additional EXAMPLES: stacks of pennies; a pile of similar size beads. Then show students a full jar of jelly beans (less than 100 jelly beans) and have students guess how many jelly beans are in the jar. Record some of the initial guesses on the board, and then have volunteers move 10 jelly beans at a time into a second jar of the same size. After each group of jelly beans is moved, students should revise their estimates. Keep track of how many groups of 10 have been removed and when the two jars have about the same amount, tell students how many groups of 10 have been moved so far. ASK: How many groups of 10 do you think are left? Why? (should be about the same because the jars look like they have the same amount of jelly beans; students might guess one more or one less) How many groups of 10 do you think were originally in the jar? (add the two numbers together, e.g = 9 groups of 10 or = 8 groups of 10) Then check together as a class how many groups of 10. ASK: How many is that? EXAMPLE: 9 groups of 10 is 90 jelly beans (check this with skip counting). EXTRA PRACTICE BLM Jelly Beans Estimating to the closest 10. PSS Mental Math, Visualization Hold up a card with dots arranged randomly. EXAMPLE:

11 Have students estimate, to the closest 10, how many dots they think there are. ASK: How many groups of 10 do you think there are? Circle a group of 10 and ask if anyone wants to change their guess. Then circle another 10 and again ask if anyone wants to change their guess. Continue in this way. ASK: Does showing a group of 10 make it easier to estimate how many there are? Finish circling all groups of 10 and discuss how grouping by 10 made it easier to count them all. Repeat with other dots on cards, but group by 5 instead of 10. Combining large dots and small dots. PSS Reflecting on what made the problem easy or hard. Draw dots randomly on a card, similar to above, but this time draw some large dots and some small dots. Have students estimate using the same method as above. Group first a group of 10 small dots, and then a group of 10 big dots. Discuss what made the number of dots harder to estimate accurately. (Because the dots are not all the same size, it s harder to get an idea for how much room 10 dots take up.) EXAMPLE: Discuss what affects estimating. Size does. Does colour? (white or dark) Pattern? (e.g. happy or sad face) Then show students these lines: ASK: Are they all the same size? (Yes) Are they easy to estimate how many? (no) Why not what makes them hard to estimate how many there are? (It s like they re different sizes because they take up so much less space one way than the other.)

12 NS2-48 Even and Odd Page xxx Goals Students will learn even and odd by pairing up groups made of even and odd objects. PRIOR KNOWLEDGE REQUIRED can count VOCABULARY pair up divide even odd equal team(s) MATERIALS several pairs of socks (see below) 8 counters for each student 8 paper circles CURRICULUM EXPECTATIONS Ontario: 2m1, 2m5, 2m7, optional WNCP: 2N2, [R, V, C, CN] Introduce even and odd numbers by pairing socks. Show the students 2 identical red socks, 3 identical blue socks and 4 identical green socks. SAY: I took these from the dryer this morning and I think I have all of them. ASK: How can I check to make sure none is missing? Have a volunteer help fold the socks in pairs. Write the word pair on the board. SAY: Socks are worn two at a time, so we pair them up when putting them away. ASK: Which colour of sock am I missing? How do you know? Pairing up faces. Draw on the board: SAY: I tried to pair up all the people, but one of them got left out. ASK: Can anyone explain what I mean by pair up? (Group people into groups of two.) Write the words pair up on the board. Give several examples of groups of happy faces and have volunteers try to pair them up. Each time, ASK: Were you able to pair them all up or was one left out?

13 Then draw on the board: PSS Generalizing from examples For each box, ASK: How many faces are there? Can you pair them all up without any left over? If I have any eight faces, no matter how they are arranged, do you think that I will always be able to pair them up? (This point is important: if we define a number as being even when you can pair up objects, it must be true no matter how they are arranged.) Pairing up eight counters and introducing even. Place eight counters on an overhead projector or stick paper circles on the board. ASK: Can I pair them up without any left over? Then pair them up. Then give students eight counters each and tell them to try to pair them up without any left over. ASK: Who was able to pair them all up? Who was not? SAY: No matter how the counters are arranged, if you have eight of them you will always be able to pair them up. Because of that, we say that eight is even. Write the word even on the board. Introduce the word odd. Draw seven happy faces on the board, arranged randomly. ASK: How many happy faces did I draw? Have a volunteer try to pair up the faces. ASK: Is seven even? If I line up the faces in a row, do you think I will be able to pair up the faces? Then try it and show students that you cannot. SAY: No matter how you arrange the seven faces, you will never be able to pair them up without any left over. So seven is not even. Numbers that are not even are called odd. Write the word odd on the board. Then draw several groups of stars on the board, and have students count the number of stars and decide whether the number is even or odd by trying to pair up the stars. Use teams to determine if a number is even or odd. Connection Real world Arrange ten counters where students can see them, using five red and five yellow counters. SAY: Let s pretend the counters are people, one team has red jerseys and the other team has yellow jerseys. Separate the red and the yellow counters and ASK: Do these two teams have the same number of players? How can I tell without counting? (pair up each red with a yellow) Demonstrate doing so. Are there an even number of people altogether? (yes) How do you know? (because we could pair up the counters) Repeat for other numbers, both odd and even. Emphasize that an even number of people can always be divided up into two equal teams an odd number of people cannot be.

14 Draw pictures on the board such as: Have students count the number of faces and decide from the picture whether that number is even or odd. Show how to pair up the faces so that you use the definition of even and odd: SAY: Because there is one left over when we pair them up, there will be one left over when we try to put them into equal teams. So 9 is odd. We could have teams that are not equal with 5 on one team and 4 on the other, but we can t have two equal teams.

15 NS2-49 Patterns with Even and Odd Page xxx Goals Students will first discover that even and odd numbers alternate and then that the ones digits of even and odd numbers can be used to identify them as even or odd. PRIOR KNOWLEDGE REQUIRED can pair up numbers can check whether a number is even or odd understands repeating patterns understands the concept of equal teams VOCABULARY even/ odd pair pair up core repeating pattern extend alternate ones digit shaded circle(d) underline before/ next MATERIALS BLM Even and Odd in a Hundreds Chart (p xxx) CURRICULUM EXPECTATIONS Ontario: 2m1, 2m5, 2m7, optional WNCP: 2N2, [R, CN, C] Look for a pattern of even and odd in consecutive numbers. Connection Patterns Do the first six questions on Workbook p. 17 together, then write on the board: Odd Even Odd Even Odd Even ASK: Do you see a pattern in whether the numbers are even or odd? Is this a repeating pattern? What is the core? (odd, even) By looking at the pattern, do you think 7 will be even or odd? Repeat for 8, 9, 10. Have students verify their prediction by drawing groups of 7, 8, 9 and 10 objects, and trying to pair them up. Connect counting by 2 to saying the even numbers. Tell students that 2 is an even number. ASK: What is the next even number? (4) And the next even number after 4? (6) Point to the odd-even pattern above and explain that to find the next even number, they skip one number and say the next. Say quietly, skip 1 then loudly, say 2, then quietly:

16 skip 3 and so on. ASK: What does this remind you of? (skip counting by 2s) Then write on the board: Have students continue writing the even numbers up to 20 in their notebooks, by using skip counting by 2s. Then have students write just the ones digits of the numbers they found: ASK: PSS Looking for a pattern Is this a repeating pattern? (yes) What is the core of the pattern? (2, 4, 6, 8, 0) Have students extend the pattern of ones digits: Connect counting by 2s from 1 to saying the odd numbers. Repeat the exercises above, this time starting with 1, to say the odd numbers. Then ASK: What are the ones digits of the odd numbers? (1, 3, 5, 7, or 9) What are the ones digits of the even numbers? (2, 4, 6, 8, or 0) Are these numbers even or odd? EXAMPLES: Bonus: Write groups of numbers on the board. EXAMPLES: 7 8 9; ; ; ; Have volunteers circle the even numbers and underline the odd numbers. Bonus: Finding the next or previous even or odd number. Emphasize that students can skip count forwards by 2s to say the next even or odd number and can skip count backwards by 2s to say the even or odd number before a given even or odd number. Is zero even or odd? Write 0 on the board. ASK: Is 0 even or odd? SAY: We cannot pair up any objects if there are no objects to pair up, so it doesn t make sense to say that 0 is even, but there isn t a leftover object, so it doesn t make sense to say that 0 is odd either. ASK: What is the ones digit of 0? (0) Does that fit with the even numbers or the odd numbers? (even) Why? (because 0 can be the ones digit of an even number, but not of an odd number, OR because the ones digits of even numbers are 2, 4, 6, 8, or 0) Now write the following pattern on the board: Odd Even Odd Even Odd Even Odd Even Odd Even ASK: What should 0 be to keep this pattern even or odd? (even) Why? (because the number next to it is odd) Explain that mathemticians call 0 even for two reasons because it fits with the repeating pattern, and because its ones digit fits with the ones digits of even numbers, not the ones digits of odd numbers. CONNECTION - Sorting BLM Even & Odd in a Hundreds Chart

17 Extensions: 1. PSS Making and investigating conjectures Have students add two odd numbers together what type of number do they always get? Start by providing examples for them (3 + 5; 7 + 7; 5 + 1; 9 + 3; 1 + 7) then allow students to investigate by creating their own examples. Bonus: Repeat with adding 3 odd numbers. Show students why this works with counters. For example, 3 counters has one extra not paired up and 5 counters has one extra so pair up the extras with each other = 8 + = 2. BLM Equal Parts guides students to discover that 0 must be even in a different way from how it was done in the lesson, this time using that even numbers are the sum of two identical numbers and odd numbers are not. 3. PSS Guessing, checking and revising, Using logical reasoning, Organizing data, Working backwards To do BLM Even and Odd in Shapes, students only need to know that 0 and 2 are even while 1 and 3 are odd. Still, the puzzle will require some thinking and will be quite challenging.

18 NS2-50 Patterns in Adding Page xxx Goals Students will discover ways to find all pairs of numbers that add to a given number. Students will use pictures and concrete materials to model addition. PRIOR KNOWLEDGE REQUIRED can draw models to add pairs of numbers can do missing addend problems VOCABULARY equal first/ last vertical line addition sentence pair MATERIALS many counters one cup for each student one large blank card per student 2-colour counters number cards (see BLM Number Cards Template (p xxx)) CURRICULUM EXPECTATIONS Ontario: 1m18, 1m25, 2m1, 2m5, 2m6, 2m7, review WNCP: 1N4, review, [V, R, CN, C] Write numbers in different ways. Write on the board: 3 + = 7. Draw seven circles, arranged randomly. ASK: How could you use the circles to find the answer? PSS Modelling (Possible answers: colour three circles and count how many are not coloured; cross out three circles and count the ones that are left; circle a group of three circles and count how many are not part of the group; and so on.) Then draw a row of seven circles: ASK: What s an easy way to choose three circles? SAY: I could choose the first three or the last three but I m going to choose the first three circles. I find it easier to remember that the first number in 3 + = 7 goes with the first circles and the second number goes with the second number of circles. ASK: How can we separate the first three circles from the others? (colour them; cross them out; circle them) Explain that these are all good ways, then show how to separate the circles by drawing a vertical line after the first three and explain that this is the model the workbook uses: ASK: Can you find the answer to 3 + = 7 using this picture? How?

19 Now tell students that you want to find all the ways of writing 7 = +. PSS Making an organized list Write 7 = + eight times on the board, all in a vertical column. ASK: What is the smallest number that can be put in the first blank? Can there be no circles before the line? (yes) Demonstrate this by drawing the line before the first circle, and write 0 in the first blank. ASK: How many circles are after the line? (7) Then finish the number sentence: 7 = Continue in this fashion by asking what is the next smallest number after 0? (1) Can there be 1 circle before the line? (yes) And so on, until all 8 number sentences are complete. Point out how the line separating the circles moves one to the right each time. ASK: Have we found all the pairs that add to 7? (yes) How do you know that we didn t miss any? (because we wrote the numbers in order) Discuss how the 8 number sentences relate to each other. PSS Looking for a pattern ASK: What number is the same in each addition sentence? (the total) How do the other numbers change each time what happens to the first number? (goes up by one) What happens to the second number? (goes down by one) Take 7 pennies and ask for a volunteer. Write on the board: Volunteer s pennies + My pennies = 7 pennies. ASK: How many pennies does the volunteer have? (0) How many do I have? (7) How many are there in total? (7). Write the number sentence (0 + 7 = 7) Now give the volunteer a penny and repeat the questions. Emphasize that you did not change the total number of pennies by giving one to the volunteer. But the number of your pennies went down by 1 and the number of their pennies went up by 1. Repeat until all pennies are transferred. Discuss how useful it is to be organized. By giving the volunteer one at a time, you made sure you didn t miss any numbers. PSS Making an organized list Write 8 = + nine times on the board, and ask students for strategies to fill in the numbers with all possible answers. Some students may suggest using a model or transferring pennies. If so, do so again. Then SAY: Notice the pattern: from one addition sentence to the next, you add one to the first number and subtract one from the second number, and you are not changing the total number. Then challenge students to find all ways of writing 6 = + without transferring pennies. Bonus: Find all ways of writing 13 = +. Activities Counters in a Cup. Students move 1 counter at a time into a cup. Students write the addition sentences based on Number in Cup + Number Not in Cup = Total Number. 2. Give each student a card to write a number sentence on. SAY: We want to find all the pairs of numbers that add to 19 (or however many students are present.) All students stand to begin. Write on the board two columns headed Number Standing and Number Sitting. Write = 19. Then have students sit down one at a time. As each student sits down, the student writes the corresponding number sentence on their card. When finished, collect all the cards and display them. Are all the addition sentences needed? SAY: PSS Using logical reasoning We know that order doesn t matter in addition. I see that some of the number sentences

20 have the same numbers. Have volunteers each erase one of each pair adding to 8 that is the same. Repeat for pairs adding to 7. Activity 3 Make a small pile of 2-colour counters. PSS Visualization Count the total number of counters together (say, 15). Then throw them up so that some land red and others land yellow. SAY: I want to know how many landed on red and how many landed on yellow. Cover them up and ASK: What are some possibilities? Write the corresponding number sentences on the board. Then uncover and count to see if the actual amounts came up in their list. CONNECTION Literature One More Bunny by Rick Walton. Students find many ways to add to numbers from 1 to 10 by using the pictures. Domino addition by Lynette Long. Students find pairs of dominoes that add to a given number.

21 NS2-51 Adding Tens and Ones Page xxx Goals Students will write numbers as a sum of 10s and 1s. PRIOR KNOWLEDGE REQUIRED can add knows the number of tens and ones in 2-digit numbers VOCABULARY ones digit tens digit sum ones card tens card MATERIALS 9 tens blocks for each student 9 ones blocks for each student BLM Hundreds Chart BLM Tens Cards BLM Ones Cards CURRICULUM EXPECTATIONS Ontario: 2m1, 2m7, 2m13 WNCP: 2N4, 2N7, [R, C] Write numbers as a sum of 10s and 1s. PSS Modelling, Looking for a pattern Provide each student with 9 tens blocks, 9 ones blocks, and a hundreds chart that fits tens and ones blocks (e.g. from BLM Hundreds Chart). Ask students to show 32 on the hundreds chart using tens and ones blocks. SAY: Each tens block represents 10 and each ones block represents 1 (count the ten ones together in one of the tens blocks), so we can write 32 = (3 tens and 2 ones). Have students continue to show various numbers using tens and ones blocks, at first with a hundreds chart and then without. Then have students write the addition sentences involving tens and ones. Finally, have students write numbers as tens and ones without using tens and ones blocks. What number am I thinking of? Have students find the number for: a) 3 tens and 4 ones (34) b) 4 tens and 3 ones (43) c) 7 tens and no ones (70) d) no tens and 4 ones (4) e) 9 tens and no ones (90) f) no tens and 9 ones (9) g) (24) h) (41) i) (60) j) (4) Break numbers into their tens and ones. ASK: How many tens are in 35? (3) What number is 3 tens? (30) How many ones are left? (5) Write 35 = Have students write various numbers as a sum of tens and ones. EXAMPLE: 42 (=40 + 2)

22 Adding tens is like adding ones. PSS Changing into a known problem Have students work in partners again and one partner has 9 ones blocks and the other has 9 tens blocks. Tell one partner to add by grouping five ones blocks with two ones blocks and finding out how many ones blocks they have altogether. Tell the other partner to add by grouping five tens blocks with two tens blocks and finding out how many tens blocks they have altogether. Ask them how many ones did the ones person have and how many tens did the tens person have. Are the answers the same? Why? Emphasize that they can find by counting the number of tens (5 + 2): = 5 tens + 2 tens = = 7 tens = 70 Repeat with several examples have students write out the tens in each number, and then see how many tens they have altogether. EXAMPLES: ; ; ; ; Now have students do similar problems without writing out the tens in each problem: = 5 tens + 4 tens = 9 tens = 90 Activity: A card trick for adding tens and ones. Photocopy BLM Tens Cards onto blue paper, once for each student, and photocopy BLM Ones Cards onto red paper, once for each pair of students. Cut them out and give each student one set of blue cards (10 to 90) and ones set of red cards (1 to 9). Show students how to add using the cards. Find the blue card 30 and the red card 4. Then place the 4 over the 0 on the tens card 30. What number do they see? (34). Repeat with various other EXAMPLES: ; ; ; Have students hold up their answers. Discuss why this works to add tens and ones. By covering up the zero with the ones digit, you are showing the number of tens beside the number of ones. This is how we write numbers. Now have students go in the other direction. Have students show the two cards that they need to make various numbers. EXAMPLES: 73 (70 and 3), 84 (80 and 4), 48 (40 and 8). Students can then write the corresponding addition sentences. EXAMPLE: 73 = Extensions: 1. Show how to subtract tens. For example, to calculate 50 20, write 50 = , then cross out 2 tens; that leaves 3 tens, so = Show how to add hundreds. SAY: Just like 10 is short for , 100 is short for The number 200 means ASK: What does 300 mean? 500? 800? Continue with adding hundreds. EXAMPLE:

23 3. Have students do the BLMs Switching Ones and Switching Tens. These sheets teach the children an application of separating the tens and ones digits to addition. It is an extension of the commutative law. For example: is the same as because = Furthermore, by switching the tens, students will see that = because = and so = For extra bonus questions, you can provide students with questions of the form: = 36 + or = Online Guide: More Extensions

24 NS2-52 Adding in Two Ways Page xxx Goals Students will use rows and columns to find the same total. PRIOR KNOWLEDGE REQUIRED understands quantity knows addition facts VOCABULARY column/ row separate different addition sentence shaded altogether total number MATERIALS connecting cubes BLM 2-cm Grid Paper BLM Hanji Puzzles (p xxx) many counters 3 toothpicks for each student CURRICULUM EXPECTATIONS Ontario: 1m18, 2m1, 2m3, 2m5, 2m13, 2m22 WNCP: 2N4, [CN, V, R] Review that two different addition sentences can represent the same number. Have students draw 2 rows of dots, each row with 7 dots. Have students separate the first row with a line between two of the dots and then write a number sentence for the model. Then have students separate the second row in a different place and write a different number sentence. Then show students how to change the two sentences into one addition sentence. EXAMPLE: = 7 and = 7 becomes = Repeat with two rows of eight dots. Bonus: Add 3 ways. EXAMPLE: = = Count by rows to write addition sentences. PSS - Modelling Ensure that all students have a clear understanding of the words row and column by asking students to identify a row and a column on a hundreds chart or the calendar. Write the words row and column on the board. Then draw the grid shown in the margin. ASK: How many squares are shaded in the first row? (Write 3 next to the first row.) How many squares are shaded in the second row? (Write 2 next to the second row.) How many squares are shaded in total? Write the sum (5).

25 Repeat by adding an additional row to the bottom. Have a volunteer count the shaded squares and write the addition sentence. Have students write the addition sentences for these grids: Count by columns to write addition sentences. Draw the same examples as above. Have students determine the number of shaded squares in each column and ask them to write the corresponding addition sentence as follows: = = = 6 Count by rows and columns to combine addition sentences. Have students compare the row sum and column sum for each drawing. ASK: Are the numbers being added always the same? (sometimes the column numbers differ from the row numbers) Are the totals still the same? (yes) Finally, combine the two ways of finding the sums (see margin) = 6 Repeat with the same examples as above. Have volunteers draw their own grids and invite others to write the corresponding addition sentences. Activity 1: Have students work in groups of 3. First, each student individually makes a 3 by 4 grid on grid paper. Assign each group a number of squares to colour, either 6, 7, or 8. Then they pass their sheet to the person to the right and that person writes the addition sentence from the row sums. Then they pass their sheet again and the next person writes the addition sentence from the column sums. All students with 6 coloured work together to make a 6 poster of all the ways they found to make a sum of 6. They cut

26 their grids and number sentences out and paste them to a common poster. Same with the groups colouring 7 or 8 squares. Compare the two models (rows of dots versus grids). PSS Reflecting on other ways to solve a problem Discuss why the grid model appears to provide more examples of addition sentences than the dots model. (The grids allows for more than 2 numbers to be added). Challenge students to find a way to use more than 2 addends with the dots model. (draw more than one line to separate the dots in separate places) Provide students with counters and toothpicks to do this concretely. EXAMPLE: = Activities Connecting cubes. Use 3 colours to write different number sentences with 3 addends. R R B B B B Y = 7 Use 2 colours to make number sentences with more than 2 addends by alternating colours. Y Y B B B Y B B = 8 Then give each student 4 red counters and 1 yellow counter. Challenge them to rearrange the counters to find many other addition sentences. Point out that the sum is always 5 because you gave them 5 counters to begin with. 3. Cooperative Cards. Play the 2-player cooperative card game described online in an At Home letter. Allow students time to discover the strategy on their own before sending the game rules home with them. Repeat the game after they learn more addition strategies. AT HOME A cooperative card game and literature connections. Extensions 1. Ask students to find 3 numbers that add to 7 in as many ways as possible without using a model. 2. BLM Hanji Puzzles 1-3. The popular Hanji puzzles invert the exercises done in class; instead of counting and adding the shaded squares in each row and column, students are given the number to be shaded in each row and column. It is easier to start by shading the full rows or columns. 3. Present the illustration shown in the margin. SAY: A student throws 3 darts. Each lands on the board. ASK: What might the total score be? ONLINE GUIDE Extension The associative law: (2 + 3) + 4 = 2 + ( 3+ 4)

27 NS2-53 Addition Strategies Page xxx Goals Students will be able to choose from a number of strategies that make adding easier. PRIOR KNOWLEDGE REQUIRED can write different models for the same sum VOCABULARY change right left first/ second opposite MATERIALS up to 20 counters for each student toothpicks a straw CURRICULUM EXPECTATIONS Ontario: 2m1, 2m2, 2m7, 2m13, 2m22 WNCP: 1N9, 2N4, 2N9, [C, V, R] Adding 1 to the first number and subtracting 1 from the second number doesn t change the sum. PSS Make an organized list Draw a row of seven dots with a line after the second dot; use a straw taped to the board as the separating line. ASK: What addition does this show? (2 + 5 = 7) Explain that there are 2 dots before the line, 5 dots after the line and 7 dots altogether. Tell students that you would like to move the line so that it shows 3 + =. ASK: Which way should I move the line left or right (or say this way or that way while pointing)? Have a volunteer move the line. Explain that you need to move it one dot to the right (this way) so that there is one more dot before the line than there was. SAY: There is now one more dot before the line how did the number of dots after the line change? (it went down by 1) Did the total number of dots go up, go down, or stay the same? (it stayed the same) PROMPT: Did we add or take away any dots by moving the line? (no) What will the new number sentence be? (3 + 4 = 7) Emphasize how the number sentence changed = 7

28 = 7 PSS Making and investigating conjectures Challenge students to predict what the new number sentence will be when you move the line one dot to the right another time. ASK: Does the first number go up by 1 or down by 1? Will there be more dots before the line or less? (there will be one more dot before the line, so the first number goes up by 1) Demonstrate this: = = 7 Continue moving the line one dot to the right, and emphasizing how the first number goes up by 1 and the second number goes down by 1. Then have students repeat the process with 8 dots. Start with: Students should record the number sentences at each stage. Practice finding another number sentence with the same answer. PSS Modelling Write across the board: = 11. SAY: If I add one to the first number and subtract one from the second number, I will still have a total of 11. Under number 6, write +1; under number 5, write -1, as shown. Complete the calculation with the new number sentence: = = = 11 Have a volunteer draw the model to show what is happening. Have volunteers continue with EXAMPLES: = 17; = 15; = 17; first writing a new number sentence and then drawing a model. Help volunteers at first by inserting +1 and -1 under the addends, but eventually have them do this step themselves. Finally, give each student up to 20 counters and a toothpick. Have them do the steps concretely and then draw them pictorially and symbolically with addition sentences. Add 1 to the second number and subtract 1 from the first number. Repeat the lesson, but this time, start by moving the line one place left in the same model and discuss how each number changes or stays the same. Doing the opposite to two numbers leaves their sum the same. Repeat the lesson, but this time, start by moving the line two or more places left or right in the same model and discuss how each number changes or stays the same. Have students change both numbers in opposite ways to make another number sentence with the same sum. Tell students how to change the first number and have them decide the correct way to change the second number, so that the sum stays the same. Students should check their answer by finding both sums.

29 Examples: = = = = = = 9 Extension: What would you take away from the third number to keep the sum the same? = 12 (Answer: = 3. Note that 5 3 = and = 12) = 12