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: 1 2 3 4 5 6 7 8 9 10

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 1. 1 2 3 4 5 6 7 8 9 10 10 20 30 40 50 60 70 80 90 100 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

EXAMPLES: 30 ____ 50; 10 _____ 30; 70 _____90; 60 ______80. 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

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 660. 3. 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: 4 + 1 + 3 + 2 + 5 + 2 + 3 = 5 + 5 + 5 + 5 = 20 (skip count 5, 10, 15, 20) 8 + 2 + 3 + 7 + 6 + 4 + 10 + 10 + 5 + 5 + 1 + 9 = 10 + 10 + 10 + 10 + 10 + 10 + 10 = 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:

10 14

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.

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

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:

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? 0 1 2 3 4 5 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: 0 1 2 3 4 5 6 0 1 2 3 4 5 6 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 10. 0 1 2 3 4 5 6 7 8 9 10 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.

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

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:

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 4 + 5 = 9 groups of 10 or 4 + 4 = 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:

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.)

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?

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.

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.

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: 1 2 3 4 5 6 7 8 9 10 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:

“skip 3” and so on. ASK: What does this remind you of? (skip counting by 2s) Then write on the board: 2 4 6 ______ ______ ______ ______ ______ ______ _____ 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: 2 4 6 ______ ______ ______ ______ ______ ______ ______ 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: 2 4 6 8 0 2 4 6 8 0 _____ _____ _____ _____ _____ 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: 13 24 87 83 90 94 Bonus: 125 876 95 431 Write groups of numbers on the board. EXAMPLES: 7 8 9; 17 18 19; 97 98 99; 43 50 67; 5 10 15 20 25. Have volunteers circle the even numbers and underline the odd numbers. Bonus: 657 789 031 8 967 540 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: 0 1 2 3 4 5 6 7 8 9 10 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

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. 3 + 5 = 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.

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?

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 = 0 + 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 1-2 1. 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 + 0 = 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

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.

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 = 10 + 10 + 10 + 1 + 1 (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) 10 + 10 + 1 + 1 + 1 + 1 (24) h) 10 + 10 + 10 + 10 + 1 (41) i) 10 + 10 + 10 + 10 + 10 + 10 (60) j) 1 + 1 + 1 + 1 (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 = 30 + 5. Have students write various numbers as a sum of tens and ones. EXAMPLE: 42 (=40 + 2)

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 5 + 2 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 50 + 20 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 50 + 20 by counting the number of tens (5 + 2): 50 + 20 = 5 tens + 2 tens = 10 + 10 + 10 + 10 + 10 + 10 + 10 = 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: 30 + 10; 40 + 20; 20 + 50; 30 + 30; 40 + 30. Now have students do similar problems without writing out the tens in each problem: 50 + 40 = 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 30 + 4 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: 20 + 7; 40 + 5; 80 + 3; 30 + 8. 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 = 70 + 3.

Extensions: 1. Show how to subtract tens. For example, to calculate 50 − 20, write 50 = 10 + 10 + 10 + 10 + 10, then cross out 2 tens; that leaves 3 tens, so 50 − 20 = 30. 2. Show how to add hundreds. SAY: Just like 10 is short for 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1, 100 is short for 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10. The number 200 means 100 + 100. ASK: What does 300 mean? 500? 800? Continue with adding hundreds. EXAMPLE: 500 + 300.

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: 13 + 5 is the same as 15 + 3 because 3 + 5 = 5 + 3. Furthermore, by switching the tens, students will see that 36 + 20 = 26 + 30 because 3 + 2 = 2 + 3 and so 30 + 20 = 20 + 30. For extra bonus questions, you can provide students with questions of the form: 46 + 32 = 36 + __ __ or 34 + 25 = 35 + __ __. Online Guide: More Extensions

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: 3 + 4 = 7 and 2 + 5 = 7 becomes 3 + 4 = 2 + 5. Repeat with two rows of eight dots. Bonus: Add 3 ways. EXAMPLE: 3 + 4 = 2 + 5 = 1 + 6 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).

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: 3 2 2 1 4 2 + 2 + 0 + 2 6 6 6 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: 2 + 2 + 2 = 6 2 + 1 + 1 + 2 = 6 2 + 2 + 2 = 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).

3 1 + 2 2 + 2 + 2 = 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

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: 2 + 5 + 5 = 3 + 3 + 3 + 3 Activities 2-3 2. Connecting cubes. Use 3 colours to write different number sentences with 3

addends. R R B B B B Y 2 + 4 + 1 = 7 Use 2 colours to make number sentences with more than 2 addends by alternating colours. Y Y B B B Y B B 2 + 3 + 1 + 2 = 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?

1 2 5 6 9 7 8

3 4 ONLINE GUIDE Extension The associative law: (2 + 3) + 4 = 2 + ( 3+ 4)

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. 2 + 5 = 7

+1 -1 3 + 4 = 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: 3 + 4 = 7 +1 -1 4 + 3 = 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: 6 + 5 = 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: 7 + 4 = 11.

6 + 5 = 11 +1 -1 7 + 4 = 11 Have a volunteer draw the model to show what is happening. Have volunteers continue with EXAMPLES: 8 + 9 = 17; 4 + 11 = 15; 5 + 12 = 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.

Examples:1. 5 + 4 = 9 5 + 4 = 9 +3 +3 -3 8 + 1 = 9 2. 5 + 4 = 9 5 + 4 = 9 -3 -3 +3 2 + 7 = 9 Extension: What would you take away from the third number to keep the sum the same? 3 + 4 + 5 = 12 (Answer: 2 + 1 = 3. Note that 5 – 3 = 2 +2 +1 and 5 + 5 + 2 = 12)

5 + 5 + ____ = 12

NS2-54 Using 10 to Add Page xxx Goals Students will use pairs of numbers that add to 10 to make adding easier. PRIOR KNOWLEDGE REQUIRED knows pairs of numbers that add to 10 can complete addition problems when one addend is missing can add 10 to a 1- or 2-digit number VOCABULARY make 10 group easier MATERIALS 20 two-colour counters or coins 3 counters for each student cards numbered 1 to 10 (3 or 4 of each number), BLM Addition Table (Ordered) (p xxx) BLM Cubes (p xxx) BLM Pass the Puck (p xxx) several shoebox lids CURRICULUM EXPECTATIONS Ontario: 1m26, 2m1, 2m3, 2m4, 2m7, 2m13, 2m22 WNCP: 1N10, 2N10, [C, R, CN] Adding 10 is easier than adding 1-digit numbers. Have students add these sums in their heads (students can use counting on their fingers): 7 + 8 8 + 9 6 + 8 5 + 6 7 + 5 Hide the answers and have students add these sums in their heads (again, students can use counting on their fingers if it helps): 10 + 5 10 + 7 10 + 4 10 + 1 10 + 2 Discuss why adding 10 is easier than adding 1-digit numbers, even though adding bigger numbers is usually harder than adding smaller numbers. PSS- Reflecting on what made a problem easy or hard. Be sure to note the pattern—to add 10 to a 1-digit number, just write the “teen” that ends with that number. Another way to see it is to use tens and ones; for example, 1 ten and 7 ones is 17. Finding an easier problem with the same answer. Now bring students’ attention to the fact that the answers in the bottom row of questions are the same as the answers in the top row (for example, 7 + 8 = 10 + 5). Then draw on the board a model for 7 + 8 by drawing a group of 7 and a group of 8. Count the two groups together to verify that this is a model for 7 + 8. Count all the dots together and then write 7 + 8 = 15.

Then draw a big circle around 10 of the dots (see margin). ASK: Now what number sentence does this show? PROMPT: How many dots are in the circle? (10) How many are not in the circle (5) How many dots are there altogether—did I change the number of dots by circling some of them? (no, there are still 15 dots) Write 10 + 5 = 15. Discuss which question is easier to answer without counting all the dots: 7 + 8 or 10 + 5? Why? (10 + 5 because you don’t even have to count; you can just look and know it’s 15) Do they have the same answer? How do you know? (yes, because we just circled a group of 10 without adding or taking away any dots) PSS – Changing into a known problem Since 10 + 5 is easier to do than 7 + 8, and we know they have the same answer, we might as well only do the easier problem. Repeat with the other four problems. ASK: How can we decide what to add to 10 to make the answers the same as the questions in the top row? PROMPT: To get 10 from 7, what do I have to do? Write 7 + ____ = 10, and have a student fill in the blank. Explain that to keep the answer the same, we need to do the opposite to 8. We added 3, so now we have to subtract 3. Write 8 – 3 = _____ and have a volunteer fill in the blank. Then write on the board: 7 + 8 = _____ +3 -3

10 + 5 = ______ Challenge students to solve these problems by subtracting 3 from the second number: 7 + 5 = 10 + _____ = ______ 7 + 7 = 10 + ______ = ______ 7 + 9 = 10 + _____ = ______ 7 + 6 = 10 + ______ = ______ Review finding numbers that add to 10. Tell students that it is important to be able to determine what makes 10 with the first number so that they can know what to subtract from the second number. Then have students practice finding what makes 10 with each number from 1 to 9. Write on the board: 10 = 1 + ___; 10 = 2 + ___. Have volunteers fill in the blanks. Continue the pattern of addition statements so that all the pairs adding to 10 are on the board. Give each student a card from 1 to 10. Write a number on the board and have all students who have the number making 10 with the number you wrote, hold up their cards. Erase the answers on the board and repeat. Picking pairs. Use a deck of cards with the face cards removed. Count to make sure you have 40 cards. Two cards “match” if they add to 10.

Finding the number that adds to 10 with the first number. Have students finish these number sentences by finding what they need to add to the first number to make 10 and then subtract it from the second number: a) 7 + 9 = 10 + _____ b) 7 + 6 = 10 + ______ c) 9 + 7 = 10 + _____ = _____ = _____ = _____ d) 8 + 6 = 10 + _____ e) 6 + 8 = 10 + ______ f) 6 + 7 = 10 + _____ = _____ = _____ = _____ d) 6 + 5 = 10 + _____ e) 9 + 6 = 10 + ______ f) 6 + 6 = 10 + _____ = _____ = _____ = _____ Making ten with the second number instead of with the first number. PSS – Reflecting on what made a problem easy or hard ASK: Which questions above have the same answer? Why did that happen? (Sample answer: 7 + 9 and 9 + 7 because you are adding the same numbers) Which question was easier: 7 + 9 = 10 + ____ or 9 + 7 = 10 + ____? Why? (for example, students might find 9 + 7 easier because it is easier to do 7 – 1 than 9 – 3) Emphasize that it doesn’t matter whether they find what makes ten with the first or the second number – they should just do what is easier. Have students add these by making 10 with the bigger number: a) 6 + 9 = 10 + ___ b) 4 + 8 = 10 + _____ c) 9 + 5 = 10 + _____ d) 7 + 6 = 10 + ____ Then try the same problems by making 10 with the smaller number. Which way is easier? (making 10 with the bigger number because then you have to subtract less) PSS – Changing into a known problem Explain that by changing one of the numbers to 10, the problem becomes an easier problem with the same answer. Changing a problem into an easier problem with the same answer is a strategy that mathematicians often use to solve problems. Adding 2-digit numbers by using tens. Then write on the board: 29 + 7. ASK: How can we make this question easier? Work through the answer with the class using the same notation as seen on Workbook p. 35. Add 1 to 29 and subtract one from 7 (see margin). Have a volunteer complete the new addition sentence: 30 + 6 = ___. SAY: Notice that 29 + 7 = 30 + 6 = 36. Repeat by adding more 2-digit numbers ending in 9 to 1-digit numbers. EXAMPLES: 39 + 5; 79 + 9; 89 + 4. 29 7 +1 -1

30 6 30 + 6 = 36 so 29 + 7 = 36. Now write on the board: 32 + 9 = 40 + ____ and 32 + 9 = ____ + 10 Discuss which way is easier to do the problem. Do they get the same answer both ways? If not, decide which is correct by counting up.

PSS – Selecting tools and strategies Now include problems that involve changing 8 to 10 (EXAMPLE: 45 + 8 = 43 + 10), and where both numbers have 2 digits (EXAMPLES: 37 + 19 or 46 + 28 or 29 + 36). Have students decide which number to change to a ten. Activities 1-2 Adapted from A Companion Resource for Grade Two Mathematics by Saskatchewan Learning. 1. Add on a 9 × 9 grid. Give each student a copy of the BLM Addition Table (Ordered) and a small counter. Have the students toss their counters and write the answers to the addition: if their counter lands on the column numbered 4 and the row numbered 9, they write the answer to 4 + 9 in that square. In this way, students randomly generate questions for themselves. 2. Make Egg Carton Dice. Make your own “dice” using BLM Cubes or have students bring in 6-pack egg cartons (bring in extras in case some students forget). Start collecting them several weeks before doing the activity. A 12-pack cut in half will also work. To make the dice, have students write different numbers in each hole in the carton or write the numbers on paper first and tape or glue them to the carton. Have them put two counters into the carton and shake. Students roll (shake) the dice and add. Make sure that when students shake the dice, they cover up any holes where coins can fall out. Students play with a partner; if they roll the same total, they get a point. Players can keep track of their scores using tallies if they are familiar with tallies and counting by 5s. To make the game harder, students can write the numbers 4, 5, 6, 7, 8, and 9 instead of 1 through 6. Students roll (shake) the dice and add. Students play with a partner; if they roll the same total, they get a point. Players can keep track of their scores using tallies if they are familiar with tallies and counting by 5s. Variations:

1. Write the numbers 4, 5, 6, 7, 8, and 9 instead of 1 through 6 on the dice. 2. Use a 12-egg carton to imitate 12-sided dice. 3. Put three coins in the egg carton to imitate rolling three dice.

3. Pass the Puck. Provide each pair of students with twelve 2-colour counters or coins (heads and tails act as two colours), a token to be the puck, and BLM Pass the Puck and a shoebox lid. Each pair of students place their common puck at the start position. Player 1 needs to toss some counters and move to an adjacent square in the next row that shows how many of each colour turned up. Players can toss their counters into a shoebox lid so that the counters do not fly across the room. Note that the start position is adjacent to all squares in the top row. Since all the adjacent squares add to 11, Player 1’s best move would be to toss 11 counters. If there is no square that they can move to, they can roll as many times as they need to. Demonstrate by “rolling” five red and two yellow, so both 5 + 2 = 7 and 2 + 5 = 7 work. ASK: Can I move if I’m on the starting point and roll this? Are there any addition sentences that add to 7 that I could move the puck to, from the starting point? (no) Why not? What do all the number sentences add to? (they all add to 11) So how many counters should I have rolled? (11) Demonstrate doing this and moving to the appropriate spot. Explain that if there is no spot they can move to, then they can toss the counters enough times so that they find a place to move to. The goal is to get the puck into the net by tossing the counters at most 10 times. Play the game once through by asking volunteers each time to tell you how many counters to roll.

Students can play the game repeatedly, taking turns who starts. The game is structured so that regardless of which player starts, the other player will finish the game. ONLINE GUIDE More Extensions Extension 1. BLM Using 10 to Add guides students to add using 10 in a different way. For example, to find 6 + 7, write 7 as 4 + something because 6 + 4 = 10. So 6 + 7 = 6 + 4 + 3 = 10 + 3 = 13. NS2-55 Using Tens and Ones to Add Page xxx Goals Students will add by separating the tens and ones by drawing tens and ones blocks and using a tens and ones chart. PRIOR KNOWLEDGE REQUIRED knows how to add tens and ones addition facts adding to 9 or less VOCABULARY tens ones altogether separate MATERIALS tens and ones blocks opaque bags CURRICULUM EXPECTATIONS Ontario: 2m1, 2m5, 2m6, 2m7, 2m26 WNCP: optional, [CN, R, V, C] NOTE: If students do not know their addition facts adding to 9 or less, they will be frustrated trying to add 2-digit numbers. Start with small tens and ones digits. Use tens and ones blocks to add 2-digit numbers without regrouping. Give students tens and ones blocks. Write “16” on the board and have students make 16 using tens and ones blocks. Tell them to set aside those blocks into a pile and then to make 13 with more blocks – students should make a separate pile. Write on the board: 16 + 13 = ____. Tell students to combine the two piles. ASK: How many tens blocks do you have? (Write “2 tens blocks” on the board.) How many ones blocks do you have? (Count them together and write “9 ones blocks” on the board.) What number do you have in total? To guide students, write on the board: 2 tens + 9 ones = _____. (29) Show how to draw tens and ones blocks to represent the numbers (one tens block and six ones blocks for 16, and one tens block and three ones blocks for 13). To

draw a tens block, draw ten small squares is an row. Students can then count the small squares to verify their addition. Emphasize that now we can count the tens and ones blocks from the drawing; we don’t need to actually have tens and ones blocks. Again, there are 2 tens and 9 ones, so 16 + 13 = 29. Count the small squares in the drawing to verify the addition. Have volunteers repeat drawing tens and ones blocks to add. EXAMPLES: 17 + 12 = ___; 11 + 16 = ___; 12 + 14 = ____; 13 + 16. PSS – Modelling Then show students an easier, but more abstract, way to draw tens blocks. Instead of drawing ten small squares in a row, draw just one long thin rectangle. Emphasize that you find it too much work to draw the ten small squares. For example, to draw a number like 32, it is much easier to draw: than Have students practice drawing several numbers using these simpler blocks. EXAMPLES: 25; 52, 37, 73. Then have students add using these simpler drawings. EXAMPLES: 24 + 33 = ___; 41 + 27 = ___. Discuss the advantages and disadvantages: it is easier to draw but harder to verify if you’re correct because you can no longer count the small squares. Activity: Give each pair of students 9 tens blocks and 9 ones blocks in an opaque bag. One partner shakes the bag and then reaches in and blindly picks out 7 blocks. The other partner takes the remaining blocks. Have students write the numbers they get individually and then together add to find the total. Students repeat the process by switching roles. Then ask volunteers to write down their addition statements they found. Discuss the results as a class. ASK: Why is everyone getting a total of 99? Why are so many addition statements different? Who had more blocks – the person who chose blindly or their partner? (the partner) Who had the greater number? Did the person with more blocks always get a bigger number? If so, challenge them to find a way so that the person with more blocks has a smaller number. If not, ASK: Why not? (a tens block has more small squares than a ones block so counting the number of blocks doesn’t tell you the number of small squares) Review separating tens and ones, then adding tens and ones. Have students show 34 with tens and ones blocks. Have a volunteer draw the tens and ones blocks on the board. Then tell students that it is convenient to group the tens and ones separately. Group the 3 tens blocks from 34 and ASK: What number does this show? (30) Group the 4 ones blocks from 34 and ASK: What number does this show? (4) Write: 34 = 30 + 4. Repeat with 52 = 50 + 2. Have students write various numbers as sums of tens and ones. EXAMPLES: 58 = ____ + ____ (50 + 8); 62 = ____ + ____ (60 + 2); 73 = ____ + ____ (70 + 3)

Then have students add tens to ones. EXAMPLES: 30 + 7 = _____; 20 + 3 = _____; Add by separating the tens and ones (no regrouping). Write on the board: 34 = 30 + 4 +52 = 50 + 2 ASK: How many tens are there in 34? In 52? Altogether? (8) Write 80 underneath the 30 + 50 and explain that 3 tens + 5 tens is 8 tens = 80. How many ones are there? (4 + 2 = 6) SAY: There are 8 tens and 6 ones. Write: 80 + 6 = ____. ASK: What number is that? (86) Repeat by asking volunteers to separate the tens and ones. Then have them add the tens and add the ones and finally combine the added tens and ones to answer the question. EXAMPLES: 43 + 54; 18 + 71; 23 + 42 + 14; 31 + 22 + 35. Bonus: Add 4 or more numbers. EXAMPLES: 21 + 23 + 31 + 14; 22 + 41 + 14 + 11 + 11. Connect the two methods. PSS – Connecting Discuss how writing 34 = 30 + 4 is the same as drawing 3 tens blocks and 4 ones blocks (the 3 tens blocks represent 30 and the 4 ones blocks represent 4) and how it is different (it is less writing to write 30 + 4 than to draw blocks) Add using a tens and ones chart without regrouping. PSS – Making a table/ chart Instead of writing 34 as 30 + 4, now write it as 3 tens + 4 ones and show students how to fill in a tens and ones chart. Explain that we can add the tens and ones separately, just as we did before.

tens ones 3 4 5 2

SAY: Now we can add the ones and tens. ASK: How many ones do we have? (6) How many tens? (8) Complete the chart by filling in the total. Have volunteers fill in additional charts with EXAMPLES: 41 + 28; 55 + 32; 73 + 22. Then write the addition question inside the chart, as shown, and have a volunteer write in the total: tens ones

4 7 1 1

Repeat with questions written inside the chart. EXAMPLES: 18 + 61; 37 + 22; 43 + 53. Conclude by writing addition questions in vertical form without a chart, but use grid paper format. Have volunteers answer the following EXAMPLES:

3 4 6 4 8 3 2 3 + 5 2 + 3 4 + 1 3 + 7 2 Then have students individually solve similar problems.

NS2-56 Regrouping Page xxx Goals Students will add two digit numbers by regrouping 10 ones for 1 ten. PRIOR KNOWLEDGE REQUIRED can decompose 2-digit numbers into tens and ones knows the tens digit as number of tens and ones digit as number of ones can find the number that makes 10 with a given number can add and subtract 10 can add single-digit numbers up to 9 + 9 VOCABULARY trade regroup ones/ tens chart MATERIALS BLM Addition Table (Ordered) (p xxx) BLM Sum Cards (p xxx) BLM Addition Table (Ordered Side) (p xxx) BLM Addition Table (Unordered) (p xxx) CURRICULUM EXPECTATIONS Ontario: 2m1, 2m6, 2m7, 2m26 WNCP: optional, [R, V, C] Addition facts to 9 + 9. Use Activity 3 from NS2-52 and Activity 1 from NS2-54 to help consolidate the addition facts up to 9 + 9 = 18. Review adding 1-digit numbers by regrouping to make 10. Write on the board: 7 + 5 = 10 + ___. Then use ones blocks to represent the numbers 7 and 5. Move part of the second pile to the first to make the number 10 as shown: Line up 10 ones blocks to show they are equal to 1 tens block. Have students work individually with additional EXAMPLES: 8 + 4 = 10 + __; 3 + 9 = __ + 10; and so on. Add 2-digit numbers using tens and ones groups by regrouping. Demonstrate 27 + 19 with tens and ones blocks on the overhead projector as follows:

27 = 20 + 7

19 = 10 + 3 + 6 30 + 10 + 6 SAY: I am just rearranging the ones blocks that I have to add together to make a pile of 10. ASK: How does that make it easier to get the final answer? What is 27 + 19? (46) Have students solve the following EXAMPLES: 27 + 38; 16 + 45; 53 + 39; 25 + 66. Add without tens and ones blocks by separating the tens and ones. EXAMPLE: 46 = 40 + 6 + 28 = 20 + 8 = 60 + 14 So 46 + 28 = 74. More EXAMPLES: 37 + 28; 14 + 47; 52 + 38; 28 + 56. Regroup tens and ones on charts. Show 3 tens blocks and 12 ones blocks and a tens and ones chart. ASK: 3 tens and 4 ones is 32—is 3 tens and 12 ones 312? (No, the number of tens and ones have to each be less than 10 to read the number this way.) PROMPT: Do we say thirty-twelve? Remind students how we count up: 30, 31, 32, …, 38, 39, 40 (not thirty-ten) Then demonstrate trading 10 ones blocks for a tens block. Have a volunteer fill out the next row of the chart with the new tens and ones. Tens Ones Tens Ones 3 12 3 12 4 2 Have students use charts to add these EXAMPLES: 20 + 26; 40 + 17; 50 + 22. Use a tens and ones chart to add 2-digit numbers. SAY: Mathematicians like to turn a harder problems, like adding 2-digit numbers, into easier ones, like adding 1-digit numbers. Display a tens and ones chart beside base ten materials again, showing the adding of the tens and ones:

Tens Ones 2 7 27 = 2 tens + 7 ones 27 1 9 +19 = 1 ten + 9 ones + 19 3 16 3 tens + 16 ones 4 6 4 tens + 6 ones Guide volunteers to fill in the appropriate boxes in the chart. Add the tens and ones first, then regroup to find the answer. Then write on the board: 54 + 28. Draw the blank chart again and ask volunteers to show 54 and 28 using tens and ones blocks. ASK: Where do I put the number of tens in 54? How many tens are there in 54? And so on. Demonstrate filling in the first two boxes. Then have volunteers fill in the remaining boxes. Repeat until all students are comfortable. EXAMPLES: 36 + 37; 42 + 19; 17 + 44. Activity 1. Solitaire. Practice single-digit addition, using BLM Addition Table 1 (Ordered) and BLM Sum Cards. Cut out all the sum cards, shuffle, and have students fill the addition table. When students have mastered this game, increase the difficulty by using BLM Addition Table 2 (Ordered Side) or BLM Addition Table 3 (Unordered). VARIATION: Set a limit on where the cards are played. The limit could be that the first card can be played anywhere, but the second card must be played in a square that touches (either a side or a corner of) a square already played. 2. Straws. Give each student a handful of straws cut in thirds. Have students bundle the straws in tens to count how many they have. Then work in pairs and write an addition sentence from totalling their straws. Students may need to regroup. Finally, work in groups of four and write another addition sentence. 3. PSS – Making an organized list, reflecting on the reasonableness of an answer This is similar to Activity 2, but give each student in a group of four, a different colour of straws, say, red, blue, yellow and green. Have students make addition sentences based on two colours. EXAMPLE: ____ red straws + ____ yellow straws = ____ straws. Tell students to make sure they have each worked with every other member of their group, so they should have a total of 6 addition sentences. Then have students make one addition sentence with all four colours and teach them how they can use this to check their pair-wise addition. First, have students line up the pairs that use the opposite colours: ____ red + ____ yellow = ____ straws ____ blue + ____ green = _____ straws ____ red + ____ blue = ____ straws ____ yellow + ____ green = ____ straws ____ red + ____ green = ____ straws ____ blue + ____ yellow = ____ straws Point students’ attention to the first row. ASK: If I add the two totals, which straws am I counting? Are there any missing? (no, none are missing because I am counting the red,

yellow, blue, and green straws) Then point students’ attention to the second row. ASK: When I count these two totals, which straws am I counting? (all of them, too!) Repeat for the third row. Emphasize that the totals in each row should be the same because you are always counting all the straws. If the two sums in each row don’t always add to the same number, they should look for their mistake! ONLINE GUIDE An extension

NS2-57 The Standard Algorithm for Addition Page xxx Goals Students will learn the standard algorithm for addition. PRIOR KNOWLEDGE REQUIRED can add tens digits and ones digits can model a number using tens and ones blocks VOCABULARY ones/ tens column regroup standard algorithim MATERIALS dice for students (or use the egg-carton dice students made) BLM Make Up Your Own Cards (p xxx) BLM Adding – Step 1 (p xxx) CURRICULUM EXPECTATIONS Ontario: 2m1, 2m3, 2m7, 2m26 WNCP: optional, 3N6, 3N8, [ME, R, C] Review adding with tens and ones charts. Include regrouping. Draw four blank tens and ones charts side by side, leaving plenty of room underneath, and have volunteers complete tens and ones charts for these questions: 35 + 47; 56 + 24; 48 + 18; 27 + 69 Introduce the standard algorithm. PSS – Looking for a pattern Show the first problem done using the standard algorithm, underneath the tens and ones chart. 1 3 5 + 4 7 8 2 Lead a class discussion. First, ensure that students understand where the numbers come from. ASK: How many ones are there in total? (12) How is that shown on the tens and ones chart? (write the 12 under the 5 and 7) How is that shown in the new way? (the 1 is written above the tens and the 2 is written under the 5 ones and 7 ones) Explain that 1 is the tens digit and 2 is the ones digit, so it makes sense to write the 1 in the tens column and the 2 in the ones column. ASK: How many tens are there in total? (8) How do you get that from the tens and ones chart—what numbers did you add? (The 3 and the 4) SAY: But 3 and 4 is only 7. How did I know to make it 8? (regroup 1 ten from the 12 ones, so add 7 + 1 = 8) How can you get the total number of tens from the new way of adding? (The 1 ten from the 12 ones is already regrouped, because it is already with the tens column, so we can add it right away: 1 + 3 + 4 = 8. Write down the standard algorithm for 56 + 24 underneath the tens and ones chart for this sum, and go through a

similar line of questioning. Then have volunteers write down the new way of adding for the remaining two sums; start them off by writing the question for them. Write down the three steps:

1. Add the ones. 2. Add the tens. 3. Regroup ten ones for a ten if necessary.

PSS – Reflecting on other ways to solve a problem Note that this new way is just combining the two steps of adding the tens and regrouping. It is just a shortcut for doing the same thing. For example, when finding the number of tens in the sum, instead of doing 3 + 4 = 7 and then 7 + 1 = 8, you can just do right away: 3 + 4 + 1 = 8. ASK: Which two additions are being combined in the second question? (5 + 2 = 7 and 7 + 1 = 8 becomes 5 + 2 + 1 = 8) Repeat for the third and fourth questions. (4 + 1 = 5 and 5 + 1 = 6 becomes 4 + 1 + 1 = 6; 2 + 6 = 8 and 8 + 1 = 9 becomes 2 + 6 + 1 = 9) Discuss why it is important to add the ones first--if they add the tens first, they may forget to include the 1 extra ten that was traded for 10 ones.) SAY: It’s a bit tricky because you have to add from right to left instead of from left to right. Many students even in grades three and four sometimes have trouble remembering to add from right to left because it is so different. That’s why it’s important to practice a lot. What if some students are having trouble? Some students may be overwhelmed by having to do both steps on Workbook p. 46. If this occurs, have them return to the first page again. This time have students do the second step on the first page by writing the number of tens in the grey box, after having just completed the first steps on all the questions. Then they can try both steps at the same time on the second worksheet. They can check their answers using the first page, since the two sheets have the same questions. EXTRA PRACTICE BLM Adding – Step 1 Estimating sums by using the closest ten. PSS – Mental math Have students guess what the closest ten is to this sum: 28 + 41. Explain that they can use what they know is the closest ten to each number being added—30 and 40. So it makes sense to guess that 28 + 41 will be close to 30 + 40. ASK: What is 30 + 40? (70) Have students find the actual sum. (69) Is 70 close to the right answer? (yes) Repeat with various sums. EXAMPLES: 19 + 32, 43 + 21, 53 + 18. ONLINE GUIDE Estimating activity Activities 1-2 1. I Have —, Who Has —?. (See NS Part 1 – Introduction) Use BLM Make Up Your Own Cards. Use 2-digit numbers on top (e.g. 28) and sums of 2-digit numbers on bottom (e.g. 17 + 25). The student with the card described would say: I have 28, who has 42? 2. Play Dominoes (See NS Part 1 – Introduction) with 2-digit numbers on one side and sums of 2-digit numbers on the other. ONLINE GUIDE Extension with BLM

NS2-58 Doubles Page xxx Goals Students will use skip counting to double numbers. Students will use the double of 5 and 10 to double other numbers. PRIOR KNOWLEDGE REQUIRED can count can skip count by 2s knows the double of 5 is 10 can add 10 to a one-digit number VOCABULARY double doubles sentence skip counting rows symmetry mirror model MATERIALS 10 counters for each student aper counters BLM What is the Double? (p xxx) BLM Doubles 1-3 (p xxx) CURRICULUM EXPECTATIONS Ontario: 1m26, 2m1, 2m22 WNCP: 1N10, 2N10, [R, V] Introduce “double.” Write the word “double” on the board. ASK: Does anyone know what the word double means? (add the same number to the number you have) If I have 3 pennies and I double the number of pennies, how many will I have? Demonstrate counting out 3 pennies and then 3 more. Explain that if you double a number, you add the same number again. Tape paper counters to the board to demonstrate this. Show 2 counters and SAY: I’m going to double my counters, so if I start with 2, I need to add 2 more. (put 2 more counters on the board) ASK: How many do I have now? Write: 4 is the double of 2. Repeat with other examples, always emphasizing the word double and using pictures or concrete objects to illustrate the doubling. Double a number by creating 2 rows of the same number. Put a row of 3 paper counters on the board and write 3 beside it. Then add another row underneath and ASK: How many are there now? Write “6 is the double of 3.” Repeat with doubling other numbers from 1 to 10. Give students ten counters. Have students write addition sentences to show the doubles of 4, 2, 5, 1, and 0.

Reading doubles from a chart. Draw 2 rows of 1, 2, 3, 4 and 5 to demonstrate doubling (see margin). 1 doubled 2 doubled … is 2 is 4 Now, using a piece of paper, cover up part of the 2 rows of 5 to show 3 doubled (emphasize that you are leaving 3 in each row uncovered): ASK: What is 3 doubled? (6) Have a volunteer use the paper to show 2 doubled, then 4, then 1, and finally 5 doubled. Now draw 2 rows of seven dots and have volunteers use it to show 3 doubled and 6 doubled, then 5 doubled, and so on. Skip counting by 2s to double. Next, demonstrate counting the number in the top row (count by 1s) and the total number (count by 2s). See margin. ASK: When I count the total number, in the two rows, how am I counting? (counting by 2s) one row: 1 2 3 … 7 two rows: 2 4 6 … 14 PSS Drawing a picture. ASK: Can you tell from this picture what the double of 4 is? Repeat for 6, 3, 7, 2, 5, and 1. Then demonstrate counting by 2s, using your fingers, to double 3—count by 2s until you have 3 fingers up. Connect to the chart above: hold up one finger at a time as you skip count and point to the number in the first row. Ask several volunteers to find the doubles of numbers up to 10 using this method. Use 5 to double. PSS Changing into a known problem. Tell students you want to double the number 8 in a different way. Show 8 as 5 coloured circles and 3 blank circles. Then double by drawing two rows. Emphasize that the 2 rows of 5 is 10 circles and the two rows of 3 is 6 circles, so 8 doubled is 10 + 6 = 16 (see margin).

8 is 5 + 3, so 8 doubled is 10 + 6 = 16.

ASK: Why is adding 10 + 6 easier than adding 8 + 8? (because adding 10 is always easy) Explain that because 10 is the double of 5, it is easier to double numbers when we split the number into 5 plus another number. Repeat the above model with additional examples. Then double some numbers using just the numbers, without the model: 6 = 5 + 1, so 6 + 6 = 10 + 2 = 12. Have students double more numbers this way.

Bonus: Find the double of 13. (use 13 = 5 + 8 or 5 + 5 + 3 to get 10 + 16 or 10 + 10 + 6 = 26) EXTRA PRACTICE BLM What’s the Double? Students double the numbers from 0 to 9 without the model. Extensions: 1. Teach students to double 2-digit numbers. See BLM Doubles 1-3. ONLINE GUIDE Details for teaching this extension 2. BLM Big Cubes and Cm (p xxx). Students discover that the number of small cubes is double the number of large cubes for any given length. Connection – Measurement 3. Challenge students to double numbers in different ways, including by subtraction, and verify that they get the same answer. EXAMPLE: 7 = 5 + 2 so 7 doubled is 10 + 4 = 14, but 7 = 10 – 3 so 7 doubled is also 20 – 6 = 14.

NS2-59 Using Doubles to Add Page xxx Goals Students will use doubles to add by using the concept of one more than or one less than. PRIOR KNOWLEDGE REQUIRED knows the doubles of numbers to at least 10 can complete number sentences where one number is missing can add 10 to a one-digit number can solve addition sentences with three addends can find one more than and one less than VOCABULARY double more than/ less than the same as two ways symmetry mirror MATERIALS counters BLM Adding with Doubles (p xxx) Curriculum Expectations Ontario: 1m26, 2m1, 2m3, 2m4, 2m7, 2m26 WNCP: 1N10, 2N10, [C, R, ME] Using one more, one less than pairs adding to easy sums. Review pairs of numbers that add to 10. Examples: 7 + ____ = 10; 6 + ____ = 10; 8 + ____ = 10 Then have students decide whether the sum is one more than or one less than 10 by comparing to two numbers that add to 10: 5 + 6 is ______one more than _______ 5 + 5 so 5 + 6 = _____ 5 + 6 is _________________________ 4 + 6 so 5 + 6 = _____ 8 + 3 is _________________________ 8 + 2 so 8 + 3 = _____ 8 + 3 is _________________________ 7 + 3 so 8 + 3 = _____ 4 + 5 is _________________________ 4 + 6 so 4 + 5 = _____ 4 + 5 is _________________________ 5 + 5 so 4 + 5 = _____ PSS – Reflecting on the reasonableness of an answer Point out that several questions were done twice. ASK: Did you get the same answer both times? Should you get the same answer both times? (yes) Discuss how useful it can be to do the same question twice – it can help you to know if you made a mistake. Now have students decide which pair of numbers adding to 10 they should use to add. EXAMPLES: 9 + 2 (one more than 8 + 2 or 9 + 1, so 9 + 2 = 11) 4 + 5 (one less than 4 + 6 or 5 + 5, so 4 + 5 = 9)

Using one more, one less than doubles. PSS – Mental math Review finding the double of numbers from 1 to 10. Then have students decide if the sum is one more or one less than a given double. 5 + 6 is _________________________ 5 + 5 so 5 + 6 = _____ 5 + 6 is _________________________ 6 + 6 so 5 + 6 = _____ Repeat with 4 + 5 (compare to both 4 + 4 and 5 + 5). Point out the questions that were done twice, and again discuss the value of doing the same question twice. Then add 6 + 7 and 7 + 6 by comparing both to 6 + 6. ASK: Do these questions have the same answer? (yes) How could you have predicted this? (they are adding the same numbers) How many more or less than a double. Show students the example 6 + 9 is 3 more than 6 + 6 = 12, so 6 + 9 = 15. OR 6 + 9 is 3 less than 9 + 9 = 18 so 6 + 9 = 15. Start by having students find the double fore using it to add. EXAMPLES: 7 + 7 = ____ so 8 + 7 = ____ 4 + 4 = ____ so 4 + 5 = ____ 9 + 9 = ____ so 9 + 8 = ____ 6 + 6 = ____ so 8 + 6 = ____ 7 + 7 = ____ so 7 + 4 = ____ 4 + 4 = ____ so 7 + 4 = ____ 3 + 3 = ____ so 3 + 5 = ____ 7 + 7 = ____ so 7 + 9 = ____ 5 + 5 = ____ so 5 + 9 = ____ 9 + 9 = ____ so 7 + 9 = ____ Bonus: 12 + 12 = ____ so 12 + 13 = ____; 14 + 14 = ___ so 14 + 11 = ____; 33 + 33 = ____ so 35 + 33 = ____; 123 + 123 = ____ so 123 + 125 = ____. ASK: Which questions were done two ways? Did you get the same answer both ways? Have students decide which double to solve before using it to add. EXAMPLES: 6 + 5 = ___ (use either 5 + 5 = 10 or 6 + 6 = 12); 7 + 6 = ___ (use either 6 + 6 = 12 or 7 + 7 = 14). 8 + 5 = ____ 9 + 7 = _____ 6 + 8 = _____ 6 + 9 = _______ Bonus: 41 + 42 = _____; 60 + 63 = _____;

312 + 310 = _____; 2341 + 2344 = _____. EXTRA PRACTICE BLM Adding with Doubles Do the opposite to both numbers to make a double. PSS – Changing into a known problem Write 3 + 5 = ____ on the board, and ASK: Does this look like a double? (no) Why not? (the numbers are not the same) Cut out circles to tape to the board and put 3 in one pile and 5 in another pile. Challenge students to move only one circle so that both piles have the same number of circles. ASK: What double do you see? (4 + 4 = 8) Did we change the total by moving one circle? (no) Explain that 3 + 5 is the same as 4 + 4, so even though it doesn’t look like a double, we can still use doubles to find it. Now explain that we are really adding 1 to the pile with 3 and removing 1 from the pile with 5. So we are doing opposite things to both piles: 3 5 +1 - 1

4 4 So 3 + 5 = 4 + 4. ASK: How can we get a double from 8 + 6? From 5 + 7? Bonus: How can you get a double from 13 + 15? = ___.

ONLINE GUIDE Activity using MIRAs Compare the different ways of adding. Write 6 + 7 = _____. Challenge the class to come up with as many different strategies as they can to solve the question. You could get them started with: Start at 6 and count on until you have 7 fingers up. Demonstrate this by saying 6 with no fingers up and then 7 with one finger up, and so on, until you have 7 fingers up. Other strategies include using 10 (6 + 7 = 10 + 3 = 13) or using doubles (6 + 6 = 12 so 6 + 7 = 13) Discuss which way is easiest? Which way is slowest? Emphasize that if students know their doubles they only have to add 1, so this is the easiest. Using 10, although easier than counting on past 6, still requires students to subtract from 7 – 4, since 4 makes 10 with 6 (or 6 – 3 if they use what makes 10 with 7). Emphasize that, by doubling, students are changing the problem into two simpler problems that they already know how to do, that is, doubling and adding 1. Choose between using 10 or using doubles. PSS Selecting tools and strategies. EXAMPLES: • 7 + 6 = ___ (This is one more than 6 + 6 and three more than 4 + 6 or 7 + 3. Some

students may think it’s easier to find pairs that add to 10 than to find doubles.) • 7 + 4 = ___ (one more than 7 + 3 or three more than 4 + 4 or 3 less than 7 + 7) • 5 + 6 = ___ (5 + 5 is both a double and a pair adding to 10, so it doesn’t matter

which way you look at this one) • 32 + 33 (using 10 doesn’t make sense; double in this case)

Online extensions for NS2-51 1. Write two numbers, one above the other, either one apart or ten apart, and have students circle the pair of digits that are different and write whether the top number is 1 more, 1 less, 10 more, or 10 less than the bottom number. Example: 26 26 is 1 less than 27. 36 36 is 10 more than 26. 27 26 More examples: 39, 49 (10 less); 68, 67 (1 more); 36, 26 (10 more); 40, 41 (1 less). Do not include pairs of numbers with both digits different (e.g. not 39, 40) 2. Ask students to write down a number that uses the same number of tens blocks as ones blocks. Have students compare their answers with other students. Did other students get the same answer or different answers? Repeat for a number that …

… uses more tens blocks than ones blocks. … uses more ones blocks than tens blocks. … uses three more tens blocks than ones blocks – ask: does 30 work here? … has a 3 in it and uses two more ones blocks than tens blocks. … uses a total of nine blocks.

If students know even and odd numbers, you can ask them to find a number that …

… is an odd number that uses more than seven tens blocks. … uses an even number of tens blocks and an odd number of ones blocks. … uses an odd number of tens blocks and an even number of ones blocks. … uses an odd number of tens blocks and an odd number of ones blocks. … uses an even number of tens blocks and an even number of ones blocks. … an odd number that uses a total of eight blocks.

4. Give students several tens and ones blocks. Have students make a triangle

using tens blocks as edges (sides) and ones blocks as vertices (corners). What number do their tens and ones blocks represent? (33) Repeat with a square (44) and a pentagon (55). Have students predict what number their blocks will show if they make a hexagon. (66)

Online Extension for NS2-52 The Associative Law. NOTE: The Associative Law for addition means that when adding three numbers, adding the sum of the first two to the third gives the same result as adding the sum of the last two to the first. EXAMPLE: 2 + 3 + 4 may be added as (2 + 3) + 4 = 5 + 4 = 9 or 2 + (3 + 4) = 2 + 7 = 9. Using connecting cubes, show students red, green, and blue cubes like this: R R G G G B ASK: How many cubes are there altogether? Then have a volunteer complete the addition sentence: 6 = _____ + ______ + ______. SAY: There are three groups of cubes, therefore we have three parts to the addition sentence. ASK: What would

happen if we put the first two groups together. What would the number sentence look like then? R R G G G B Have a volunteer finish the number sentence: 6 = ___ + ____. Then, pull the red and green cubes apart, and put them back into the original set. ASK: Does it makes sense to write: 2 + 3 + 1 = 5 + 1? Is the sum on the left the same as the sum on the right? Does it make sense to put an = sign between them? (yes) Did I change the total number of cubes by moving the cube in the second group to the first group? (no) Do you think you would change the number of cubes if you moved the second group to the third group? What would number sentence look like? Have a volunteer move the green cubes to join the blue cubes instead and write the number sentence on the board. Then write on the board: 6 = 5 + 1 6 = 2 + 3 + 1 6 = 2 + 4 Repeat with several examples. Always make the connection between the connecting cubes and the addition sentences. Encourage students to write the longer number sentence. EXAMPLE: 2 + 3 + 1 = 5 + 1 = 2 + 4. Circle the grouped numbers to help them: 2 + 3 + 1 = 2 + 3 + 1 = 2 + 3 + 1 = 2 + 3 + 1 2 + 3 + 1 = 2 + ____ = ____ + 1 = ______ Then draw several examples of “piles” of circles on the board. SAY: Imagine that the circles are counters in piles. We can move the first two piles together or the second two piles together. Demonstrate moving actual counters at the same time as drawing the circles symbolically on the board. Using ten counters, have students work in pairs to create a matching addition statement with three addends. Then, have students regroup the counters to form two new two-addend addition sentences which still have a sum of ten. Have one partner write the number sentence obtained by grouping the first two piles together and the other partner group the last two piles together. Challenge students to use the sums 11, 12, 13, 14 … 18 to create addition sentences which follow the Associative Law. Provide students with BLM The Associative Law (p xxx). Ensure that they understand that the middle pile is being moved to either the first pile or last pile. Then pull all the piles together to find the total. Practice drawing what the piles would look like after being moved and writing the addition sentence. Online extensions for NS2-54 2. Which is quicker: the brain or a calculator? Challenge students to add faster than a calculator. First give students problems that only add to either 5 or 10, and tell them you are doing so. You solve all the problems by punching them onto a calculator at normal speed, and the students solve all the problems in their head. Did they finish ahead of you? (Only they need to know the

answer; do not ask students to tell the rest of the class whether or not they finished ahead of you) Then progress to some problems adding to 4, some to 5, and some to 6. Teach students to determine what makes 5 with the first number and then decide whether the second number is one less than, equal to, or one more than that number. Repeat with problems that add to 3, 5, or 7 with 2 less than and 2 more than. Then progress to problems that add to 9, 10, or 11, and then to 8, 10, or 12. Finally combine two or more types of problems on the same handout. 3. Show students the following puzzle: SAY: Suppose we have: 2 4 5 Explain that by drawing arrows on some of the lines you are showing which numbers are to be moved from the squares into the circle to be added. The following examples indicate how each number from the diagram above would move into the circle. 2 4 5 Draw 2 or 3 arrows to indicate that those 2 or 3 numbers are to be added and guide students to find the answers: 6 11 (ANSWERS: 2 + 4 = 6 and 2 + 4 + 5 = 11) Then challenge students to find: (ANSWERS: 2 + 5 = 7 and 4 + 5 = 9) Repeat with EXAMPLES: 1 2 3

2 3 4 3 5 6 Then draw the following and ASK: What do you think this means? 1 2 3 ANSWERS: 1 + 1 3 + 3 2 + 2 Have students compare the answers to the following questions for each example above: 1. + and + (The answers are 2 + 5 = 7 and 3 + 4 = 7) The pictures for both answers are the same: 2. + + 4 + 4 = 8 5 + 3 = 8 answer picture 3. + + 6 + 3 = 9 4 + 5 = 9 answer picture Students will see that the numbers are always the same and the pictures for the answers should help them see why. BLM Addition Puzzles 1-3 (p xxx) provides practice with this type of puzzle.

Online Extension for NS2-56 A different way to use tens and ones charts to add. SAY: I will still add the tens and ones separately, but I am going to put the tens and ones in a different place this time. To make it easier, let’s just start by adding the ones. We’ll add the tens later. Draw a tens and ones chart on the board as follows: Tens Ones 2 7 1 9 1 6 ones 7 + 9 = 16 SAY: Remember, we are just considering the ones at this point. ASK: Why am I writing 1 in the tens column? Why do I have a 6 in the ones column? (7 + 9 = 16; 10 ones = 1 ten and is written in the tens column leaving 6 ones in the ones column) Repeat with examples, asking students to add only the ones. Emphasize why you are putting the tens digit in the tens column and the ones digit in the ones column. Then extend the chart so that you can add the tens as well. Tens Ones 2 7 1 9 1 6 ones 3 tens ASK: Why am I writing 3 tens in the same column as the 1 ten from the 16? Can you tell easily how many tens are in 27 + 19 from this chart? Can you tell easily how many more ones there are in 27 + 19? What is 27 + 19? Do more examples. Have students find the ones first, then the tens and then the total. Bring them to the point where they do not need the tens and ones charts to do the adding. To start, for EXAMPLE:

Tens Ones 2 7 + 1 9

1 6

3

4 6

Eventually move away from writing the tens and ones on top. NOTE: This method can provide a good intermediary step before learning the standard algorithm. It is important, however, not to replace it with the standard algorithm, as it will not always work so efficiently with 3-digit numbers: 2 3 7 2 3 7

+ 4 6 5

+ 4 6 5

1 2 1 2

9 9

6 6

6 10 2 2

1 0

6

7 0 2

Notice that the algorithm used for 2-digits does not quite work for three digits when you carry a 1 and the two tens digits add to 9. You need to add a bit of inefficiency to make it work, as indicated above. Nonetheless, even when adding 3-digit numbers, some students may prefer this method as a starting point. Online Extension for NS2-57 2. Provide BLM More Addition Puzzles 1-5 (p xxx). The last page provides blank puzzles. The following are sets of 4 numbers that you could put in the squares for students (or allow students to choose their own): 2,3,6,4; 6,7,5,9; 18,7,6,15; 27,34,18,16; 23,16,17,35. Discuss the self-checking mechanism this provides. Online Activity for NS2-57 Estimating Game. Students need the egg carton dice described in NS2-54: Using 10 to Add for this game. Draw on the board the picture in the margin.

Give each student four tokens to mimic rolling 4 dice in the egg carton. Students try to place the numbers in the four boxes to obtain a number as close as possible to 100, and record their answer. Repeat, using the same numbers, to obtain a number as close as possible to 70, and then finally to 40. Students could use BLM Estimating Game, which provides the outline for the boxes. Variation: Students roll one die (or use one token) at a time. Students record the number in a square after each roll and are not allowed to change their minds based on their next roll. Online Details for teaching Extension 1 of NS2-58 First, use ten to double in the same way we used 5 to double on Workbook p. 49. Example: To double 13, fill in the blank: 13 = 10 + ____. Then double the 10 and the number in the blank: 13 = 10 + 3, so 13 doubled is 20 + 6 = 26. Emphasize that this is the same answer as in the bonus above. No matter how you double 13, you still get the same answer. Have students practice using 10 to double various numbers. EXAMPLES: 11, 14, 13. Bonus: 17, 18, 16, 19. Next, double tens by doubling the first digit. 20 is two rows of 10, so 20 doubled is four rows of ten. This means that 20 doubled is 40. Have students double 30 (60) and 40 (80). Bonus: Double 70 (140) and 50 (100). Then double numbers by separating the tens and ones: 34 = 30 + 4, so 34 doubled is 60 + 8 = 68. Repeat with more 2-digit numbers where both digits are less than 5. EXAMPLES: 24, 21, 32, 33, 31, 42, 41, 43, 44, 34, 22, 23.

Online Activity for NS2-59 Explain that we see doubles in a mirror. We see the real object and then we see it again in the mirror. Give students MIRAs. Tell students to place four counters in front of the MIRA. Ask a volunteer to draw what they see. Have another volunteer write an addition sentence based on the number of counters on each side of the MIRA and the number of counters they see altogether. ASK: Do you see a double? Using MIRAs, have students model doubles plus one, such as 4 + 4 + 1, using counters. One counter will always be outside the range of the mirror, as shown. Provide students with BLM Doubles and Mirrors.