unit 7 number sense: addition and subtraction with … for new... · · 2017-06-15copyright ©...

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Number Sense H-1

IntroductionIn this unit, students will review counting and ordering numbers to 100. They will also explore various strategies and tools for solving addition and subtraction problems within 100. Students will use hundreds charts, base ten blocks, ten-frames, and number lines. Strategies will include counting forwards and counting backwards, and making use of the connection between addition and subtraction.

Meeting Your Curriculum

Alberta—All lessons in this unit are required, except Lesson NS2-26. Lesson NS2-26 is recommended, as it reviews Grade 1 material that is used in Lesson NS2-27.

British Columbia—All lessons in this unit are required, except Lesson NS2-26. Lesson NS2-26 is recommended, as it covers material that is used in lesson NS2-27.

Manitoba—All lessons in this unit are required, except Lesson NS2-26. Lesson NS2-26 is recommended, as it reviews Grade 1 material that is used in Lesson NS2-27.

Ontario—All lessons in this unit are required, except Lesson NS2-26. Lesson NS2-26 is recommended, as it reviews Grade 1 material that is used in Lesson NS2-27.

Recurring Games

The following games and activities recur throughout this unit. Rules and materials vary per lesson.

I Have —, Who Has —? Each student needs one card to play (see sample below). You can make the cards or have students make them using BLM Game Cards (p M-2). The blank spaces at the top and bottom of each card can be filled with numerals or representations of numbers: an arrangement of dots, tens blocks, an addition or subtraction sentence. The student with the card shown below would start by saying, “I have 3. Who has 7?” The students who has 7 on top would respond with, “I have 7. Who has [whatever is on the bottom of the card]?” and so on.

I have

3

Who has

Unit 7 Number Sense: Addition and Subtraction with Numbers to 100

CA 2.1 TG Unit 7 p1-44 V4.indd 1 27/04/2017 11:02:04 AM

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H-2 Teacher’s Guide for Grade 2

Catch. You will need a small ball or paper object that students can catch. Toss the ball to a student while saying a number. The student catches the ball and repeats the number. The student then tosses the ball back to you and says whatever “next” number you have asked for (e.g., the next number counting backwards. Ensure that everyone gets a chance to play.

Materials

Hundreds charts and base ten materials. Make a copy of BLM Hundreds Chart (p M-5) for each student, and laminate it if possible. Use additional photocopies of this BLM as required. Students will often use this hundreds chart with 1 cm connecting cubes and tens and ones blocks. If you do not have such cubes or blocks, or if your students need larger manipulatives, they can use BLM Hundreds Chart—Five Rows (p H-47) with paper ones and tens blocks from BLM Base Ten Materials (p M-4). Copy and laminate as many tens and ones blocks as required. Also available: a slightly larger hundreds chart on BLM A Larger Hundreds Chart (p M-13).

A hundreds chart for whole-class teaching. For whole-class discussions and demonstrations, you will need a large hundreds chart. You might draw one on the board, or project a hundreds chart transparency.

Tens and ones blocks. You will often need tens and ones blocks. Two different colours of blocks is ideal for demonstrating addition (e.g., 3 red blocks + 4 blue blocks is 7 blocks altogether.)

As an alternative, you can use 1 cm connecting cubes, and have students link ten together to create a tens block. If you don’t have 1 cm connecting cubes or tens and ones blocks, you can use BLM Base Ten Materials to make some. Photocopy the BLM onto red and blue paper, glue it to bristol board or thin cardboard (e.g., a cereal box), and cut out the materials for your students. Be aware, however, that many students will find these thin blocks hard to manipulate.

Long number lines. You might wish to provide students with long number lines from 0 to 100 for some of the lessons in this unit. Although metre sticks or measuring tapes can be used, you can also make a number line using a hundreds chart. Cut out a hundreds chart (you can use BLM Hundreds Chart or BLM A Larger Hundreds Chart) leaving extra space to the left of the chart. Fold the chart to make a cylinder and tape it together so that when the first row ends, the second row starts. Cut out the rows in one long spiral starting underneath the 1; this will form one long strip with the numbers in order from 1 to 100. You can make the number line yourself, or make the cylinders and have students cut them.

Two-colour counters. Two-colour counters are called for in this unit. If you do not have these, you can make your own using dried beans by painting one side of the beans a different colour. As an alternative to two-colour counters, you could put two colours of connecting cubes or blocks into an opaque bag. Instead of tossing a certain number of two-colour counters, students draw a handful of cubes out of the bag without looking.

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100


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Number Sense H-3

Shoeboxes to contain tosses of two-colour counters. It can be useful to have a container to prevent the tosses of two-colour counters from flying onto the ground when students use them. Shoeboxes or shoebox lids work well for this purpose.

Generic BLMs. In addition to the BLMs found at the end of this unit, the following Generic BLMs, found in section M, are also used in Unit 7:

BLM Base Ten Materials (p M-4) BLM Hundreds Chart (p M-5) BLM Game Cards (p M-2) BLM Hundreds Chart—One Row (p M-6) BLM A Larger Hundreds Chart (p M-13) BLM 1 cm Grid Paper (p M-7)

Assessment. The assessment checklist for this unit can be found in section N. The following table indicates the lessons covered by a test, which can be found in section O.

Test Lessons NS2-20 to 25, 27



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GoalsStudents will count orally to 100. Students will check their counting

and identify mistakes in counting by keeping track.

PRIOR KNOWLEDGE REQUIRED

Can count onCan count to 20

MATERIALS

ball or paper objectlarge hundreds chart for demonstrationBLM Hundreds Chart (p M-5, optional)tens and ones blocks for demonstration

Count orally to 30. Review counting to 20. Then write the numbers from 20 to 30 on the board, point to each number, and say it aloud. Repeat, but this time emphasize the last part of the word while underlining the ones digit. ASK: Which two numbers end with the same digit? (20 and 30 both end with 0) Do the words for the numbers sound the same in any way? (they both end with a “tee” sound) Look at the other numbers, from 21 to 29—what are the last digits? (1 to 9) On the board, write 30 and 13. Have students listen carefully while you say the numbers, then ASK: What part sounds the same? (“thir” or “thirt” or “thirtee”) Emphasize that for numbers in the “teens”—thirteen, fourteen, fifteen, and so on—we hear the last digit first. Point to and say “thirteen”: we hear a sound that’s close to “three” and then “teen.” SAY: It’s the same for the number fourteen—we hear “four” and then “teen”—for fifteen, and so on. However, the pattern changes for numbers in the 20s. First we hear a sound that’s close to “two” and then the ones digit. Say the numbers again to demonstrate.

Continue counting orally to 100. Write 31 on the board and say it aloud. Then write 32. ASK: How would you say this? Continue through the 30s, first in numerical order and then in random order. Repeat with the 40s, 50s, ... , 90s. Emphasize the connection between how we say 40 and 14, 50 and 15, and so on (just take the “n” sound off “teen” to get the other number). Ask students to say 20, 30, 40, 50, 60, 70, 80, and 90, first in numerical order and then in random order.

ACTIVITY 1 (Essential)

1. Catch. (see unit introduction) Say any number less than 100 and ask students to say the next number. Do not include numbers ending in 9.

CURRICULUM REQUIREMENTAB: requiredBC: requiredMB: requiredON: required

VOCABULARYnumbers to 100ten-frame

NS2-18 Counting to 100Pages 102–103


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Number Sense 2-18 H-5

The number after numbers ending in 9. SAY: The numbers you say after 29, 39, and 49 are the hardest to remember. Once you remember that 30 comes after 29, you can easily count to 39 (count from 30 to 39 together). It’s remembering what comes after 39 that’s hard. Look at a hundreds chart together or give one to each student (see BLM Hundreds Chart). Point to the 2 and 3 in the 20s and 30s. SAY: We know the 20s start with a sound that’s close to “two” (“tw”). The thirties start with a sound that’s close to “three” (“th”); remember how it sounds more like 13 than 3 (take off the “n” sound). ASK: What comes after 3? (4) What should the number after the 30s sound like? (forty) (PROMPT: Think of the number that comes after 13, but take off the “n” sound. Then chant the numbers from 40 to 49 as a class. Continue to 100.

We count by grouping in tens. Tell students that you heard someone count like this: “one, two, … , twenty, … , twenty-nine, twenty-ten, twenty-eleven, … , twenty-twenty.” ASK: Is this right? What’s different about this counting? Why do you think the person counted like this? (they were grouping numbers in groups of twenty) Explain that in English we start counting over at ten and groups of ten numbers sound the same. That’s why numbers twenty to twenty-nine have twenty in common; thirty to thirty-nine have thirty in common, and so on.

ACTIVITY 2 (Optional), ACTIVITY 3 (Essential)

2. Have students stand in a line. The first person in line says “one,” the next person says “two,” and so on to 100, with one catch: any student who says a word that has the sound “four” in it (EXAMPLES: 14, 24, 40–49) has to move to the front of the line. (EXAMPLE: I say 14 and move; the student who stood behind me before I moved says 15.) Repeat with the sound “five.” (NOTE: 50–54 and 56–59 have a “fif” sound, not a “five” sound.) VARIATION: Students stand in a circle and whisper the next number to the next person; special numbers are said out loud.

3. Catch. (see unit introduction) Say any number less than 100 and ask students to say the next number. Include two-digit numbers ending in 9.

Say and write two-digit numbers. Display a large hundreds chart and ask students to first say, and then write, numbers as you point to them. Point to numbers one after the other that look or sound similar.

Say the number given groups of tens and ones. Draw or present various objects grouped by 10s and 1s (e.g., crayons or dots). Have students say what number is represented. Point out that counting in this way, where objects are grouped by 10s and 1s, is like counting using tens and ones blocks. Show a couple of examples using tens and ones blocks. Repeat using ten-frames.

NOTE: Some languages group numbers differently.

EXAMPLES: 23, 32; 6, 16, 60; 65, 55, 95



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Extensions1. Teach students to count to 200, or even 1000.

2. Which is longer? Measure a chain of 100 paper clips against the height of one school floor. Students can hang the chain from the top of a stairway.

3. Count by keeping track. Remind students how to count and keep track of all the letters in a sentence, as they did in Unit 2, but use longer words and sentences. EXAMPLES (see below for subtotals):

1. Sara likes to jump rope with her sister. 2. John likes to bake chocolate chip cookies with his father. 3. Matt likes to jump rope with his brother and bake chocolate chip

cookies with his mother.

SUBTOTALS: 1. 4, 9, 11, 15, 19, 23, 26, 32 2. 4, 9, 11, 15, 24, 28, 35, 39, 42, 48 3. 4, 9, 11, 15, 19, 23, 26, 33, 36, 40, 49, 53, 60

Students can compare their answers. As before, discuss how counting letters in this way gives students an opportunity to check their work and find mistakes.

Have students complete BLM Counting and Colours (p H-45).

NOTE: Extensions 4–6 should be done in order.

4. Number words for multiples of 10. Teach students to read the number words for multiples of ten, up to one hundred. First review the number words from one to twenty. Then focus on “ten” and “twenty.” Write these two words on the board and ask what number each word represents. (10 and 20) Write “10” above “ten,” and “20” above “twenty.” Then write “thirty” on the board. Have the class read the word, and write “30” above “thirty.” Ask students which number words the word “thirty” reminds them of. (three, thirteen) Ask students which letters these words have in common (th, or thirt) and what is the same about all of the numbers. (all the numbers have a 3) Repeat with “forty,” “fifty,” “sixty,” “seventy,” “eighty,” and “ninety.” Then write “one hundred” on the board. Have students read the word, and then write “100” above it. Ask students which smaller number word they recognize from “one hundred.” (one) Ask students where they see one (1) in the number 100. (the first digit)

Write the numeral above the word.

a) thirty b) ten c) eighty d) twenty

e) ninety f ) seventy g) forty h) sixty

i ) fifty j ) one hundred

Answers: a) 30, b) 10, c) 80, d) 20, e) 90, f) 70, g) 40, h) 60, i) 50, j) 100

CONNECTION

Art—On BLM Counting and Colours, students discover how mixing colours can make brown.


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5. Number words up to 100. Teach students to read any number word from 20 to 100. Write on the board the number words from “twenty-one” up to “twenty-nine.” Read the words as a class, and write the matching numerals above the number words. Guide students to see the connection between the number words and the numerals. For example, for twenty-four, SAY: The “twenty” part means we write “2” and the “four” means we write “4.” Point out that there is always a hyphen (-) between the word “twenty” and the word for the one-digit number. Repeat with “thirty-one” to “thirty-nine,” and so on, up to “ninety-nine.”

Write the numeral above the word.

a) thirty-seven b) fifty-three c) thirty-five

d) twenty-nine e) ninety-six f ) seventy-two

g) forty-one h) sixty-six i ) fifty-eight

j ) forty-four k) eighty-five l ) ninety

m) thirty-four n) fifty o) eighty-eight

p) ninety-nine

Answers: a) 37, b) 53, c) 35, f ) 29, e) 96, f ) 72, g) 41, h) 66, i ) 58, j ) 44, k) 85, l ) 90, m) 34, n) 50, o) 88, p) 99

6. I Have —, Who Has —? (see unit introduction) Make game cards using BLM Game Cards (p M-2) with numerals on the top and number words on the bottom. Use numbers from 20 to 100.



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NS2-19 Hundreds ChartsPage 104

GoalsStudents will use patterns to find numbers on a hundreds chart.


Can count to 10 using a hundreds chart

MATERIALS

10 tokens or coins per studentBLM Hundreds Chart—One Row (p M-6)BLM A Larger Hundreds Chart (p M-13)BLM Hundreds Chart (p M-5)BLM Hundreds Chart Pieces (p H-46)number cards to 100 (optional)

Review finding numbers in the first row of a hundreds chart. Give each student 10 tokens and a large strip of paper with the first row of a hundreds chart (e.g., from BLM Hundreds Chart—One Row). Ask students to place a token on each number from 1 to 10 as you say the numbers in random order. Stop when the row is full.

Review finding numbers in the second row of a hundreds chart. Draw the first two rows of a hundreds chart on the board and review how to find numbers in the second row using the first row as a guide, e.g., to find 17, find 7 and move down a row. Have volunteers use this method to find various numbers in the second row. Hand out BLM A Larger Hundreds Chart. Have students find and lightly colour the first two rows. Ensure that all students colour the correct rows. Call out the numbers in the exercises below, and check that students are covering the correct numbers. NOTE: You might have students clear the tokens off the hundreds chart after completing part a) to make part b) slightly more challenging.

Exercises: Place a counter on the numbers.

a) 17, 14, 9, 16, 18

b) 12, 15, 7, 20, 13

Find numbers on the entire hundreds chart. Tell students to look at the third row. ASK: How can we find 27 if we know where 7 is? (find 7, then move down until you find 27) How can we find 57 if we know where 7 is? (move down from 7 until you find the 57) Repeat with various numbers, including numbers that end in 0. Once again, call out the numbers in the exercises below, and check that students are covering the correct numbers.


VOCABULARYhundreds chart


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Exercises: Place a counter on the numbers.

a) 35, 75, 95, 85, 15, 65, 25, 55

b) 17, 67, 87, 97, 47, 57

c) 30, 60, 50, 90, 10, 70, 80, 20

d) 53, 46, 81, 42, 75, 90, 45, 33, 77

Use the hundreds chart to find the next number, the previous number, and the number in between. Tell students to find the number 37 and place a token on that square. ASK: What is the next number? (38) Write “37 ” on the board. Have a volunteer fill in the blank. Repeat with various numbers. Have students write the number you say and the number that comes next in their notebooks. Repeat with numbers that come before a given number, and then with numbers that come in between two given numbers.

Exercises: Use a hundreds chart to find the numbers.

1. Write the number that comes after.

a) 47 b) 51 c) 79 d) 90

Answers: a) 48, b) 52, c) 80, d) 91

2. Write the number that comes before.

a) 58 b) 43 c) 80 d) 71

Answers: a) 57, b) 42, c) 79, d) 70

3. Write the number that is in between.

a) 38 40 b) 53 55 c) 68 70 d) 89 91

Answers: a) 39, b) 54, c) 69, d) 90

Find groups in a hundreds chart. Arrange cut-out “pieces” of a hundreds chart on the board. Label each piece with a colour. Have students find and colour the pieces on BLM Hundreds Chart. EXAMPLES:

Blue

12 13

22 23

Red

27 28 29

37 38 39

Yellow

35

45

55

Green

41 42

51 52

61 62

71 72

Purple

67 68 69 70

77 78 79 80

87 88 89 90

Orange

92 93 94

ACTIVITY (Optional)

Students cut out the pieces from BLM Hundreds Chart Pieces and glue them in the correct place on BLM A Larger Hundreds Chart.



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Find missing numbers in a hundreds chart. Create part of a hundreds chart with cards. Withhold three cards. See the margin for an example.

Give each missing card to a volunteer to put in the correct place. Remove different cards from the chart, shuffle them, and have volunteers put them back. Repeat with a different part of the hundreds chart. Remove more and more cards from the chart. Finally, have volunteers write the missing numbers in the empty spaces instead of referring to the cards. Eventually, you should have no cards on the board—only numbers written in by students.

NOTE: Instead of using cards you can draw a piece of a hundreds chart on the board leaving some squares blank.

Extensions1. Give students BLM Hundreds Chart—Five Rows (p H-47). Have

students use the hundreds chart to add: 5 + 3, 15 + 3, 25 + 3 (EXAMPLE: shade 5 and circle the next 3 numbers). ASK: What pattern is in the answers? Can you predict 65 + 3? 75 + 3? 85 + 3? Verify the prediction on a large hundreds chart. Repeat with 19 + 3, 29 + 3, 39 + 3.

2. The reading pattern in Japanese is top to bottom and then right to left. Show students the phrase “Once upon a time” written using the Japanese reading pattern (see margin).

Together, fill out a blank hundreds chart using this reading pattern.

Answer: 91 81 71 61 51 41 31 21 11 1

92 82 72 62 52 42 32 22 12 2

93 83 73 63 53 43 33 23 13 3

94 84 74 64 54 44 34 24 14 4

95 85 75 65 55 45 35 25 15 5

96 86 76 66 56 46 36 26 16 6

97 87 77 67 57 47 37 27 17 7

98 88 78 68 58 48 38 28 18 8

99 89 79 69 59 49 39 29 19 9

100 90 80 70 60 50 40 30 20 10

3. Have students make their own hundreds chart, using grid paper or BLM 1 cm Grid Paper (p M-7) that has three errors—in other words, three squares in the chart have an incorrect number. Students exchange charts with a partner and try to find the errors in their partner’s chart.

15 16 18 19

25 26 27 29

35 37 38 39

45 46 47 48 49

m a u O

e p n

t o c

i n e


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NS2-20 More Tens and Ones BlocksPages 105–107

GoalsStudents will use tens and ones blocks to represent numbers and

to find numbers on a hundreds chart.


Can find a given number on a hundreds chart

MATERIALS

ones and tens blocksBLM Hundreds Chart (p M-5)overhead projectortransparency of BLM Hundreds Chart (p M-5)BLM Hundreds Chart and Base Ten Materials (p H-48)BLM Game Cards (p M-2)BLM 1 cm Grid Paper (p M-7)

Count past 20 using the hundreds chart. Review counting to 20 using the first two rows of a hundreds chart. Then give each student at least 40 ones blocks and BLM Hundreds Chart. Have students count their blocks by using the chart. ASK: How many blocks did you count? How many full rows did your blocks cover? How many blocks in the next row did you need? Record answers on the board. (EXAMPLE: 35 blocks, 3 rows and 5 more blocks) Have students predict how many full rows they will fill and how many more blocks they will use to make these numbers: 28, 32, 23, 31, 13, 30, 36. Verify their predictions on an enlarged hundreds chart.

Exercises

1. Guess how many rows and how many extra squares the number will fill.

a) 33 full rows b) 29 full rows 33 extra squares 29 extra squares

c) 25 full rows d) 38 full rows 25 extra squares 38 extra squares

Answers: a) 3, 3; b) 2, 9; c) 2, 5; d) 3, 8

2. Check your guesses in Exercise 1 using ones blocks and a hundreds chart.

Tens blocks. ASK: How many full rows and how many more blocks do we need to show 74? Record the students’ predictions. To verify the predictions, begin placing ones blocks in order on a transparency of BLM Hundreds Chart on an overhead projector. After you finish a few rows, SAY: I’m tired of placing so many ones blocks in order. ASK: Does anyone


VOCABULARYones digittens digit



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remember what we used before to cover an entire row? (a tens block) Show students a tens block. Count the individual ones blocks visible within a tens block to verify that there are ten. Then cover one full row with a tens block. ASK: Do we need to cover another full row or is 74 in the next row? (repeat until 74 is in the next row) How many ones blocks do we need in the next row? (4) How many full rows did we cover? (7) Record the answer on the board: 74 is 7 full rows and 4 more blocks. Repeat with various numbers, using tens blocks for full rows.

Find numbers on a hundreds chart. SAY: How many full rows do I have to cover before I get to 63? (6) Where is 63 in the next row? (the third one) Count six full rows using tens blocks and then count three in the next row using ones blocks to demonstrate finding 63. Invite volunteers to find various numbers, then have students find numbers on their own hundreds chart.

ACTIVITY 1 (Optional)

1. Assign each student a number up to 49. Students display their number on a hundreds chart by cutting the correct blocks from BLM Hundreds Chart and Base Ten Materials.

Compare two methods of finding numbers on a hundreds chart. Compare the first method students learned with the method they learned in this lesson. Use the number 45 as an example:

•Find5inthefirstrow,thenmovedownuntilyoufind45.

•Movedownorcoverfourfullrowsandthencountacrossfive squares.

Point out that you’re really doing the same two steps but in different order. Whether you move across then down, or down then across, you end up in the same place.

Show numbers using tens and ones blocks without a chart. SAY: You can use blocks without the hundreds chart to represent a number. Show students 3 tens blocks and 7 ones blocks. Draw a T-chart on the board and label the columns “tens” and “ones.” ASK: How many tens blocks do I have? (write 3 in the tens column) Repeat for ones blocks and the ones column. ASK: If we placed these on the hundreds chart, what number would we get? (37) Check by counting each cube, including the 10 in each tens block. Then place the blocks on a hundreds chart and emphasize that 37 is the last square covered. Repeat with various numbers, this time having students fill in the chart and write the number.

For the following exercises, give each student 9 tens blocks and 9 ones blocks.


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Exercises: Show the number using tens blocks and ones blocks. Write how many tens blocks and ones blocks you used.

a) 47 tens blocks b) 59 tens blocks 47 ones blocks 59 ones blocks

c) 62 tens blocks d) 74 tens blocks 62 ones blocks 74 ones blocks

e) 89 tens blocks f ) 98 tens blocks 89 ones blocks 98 ones blocks

Answers: a) 4, 7; b) 5, 9; c) 6, 2; d) 7, 4; e) 8, 9; f ) 9, 8

Tens digits and ones digits. Write “27” on the board. ASK: Which digit shows me the number of tens blocks I need to make 27—the 2 or the 7? (the 2) Which digit shows me the number of ones blocks I need to make 27—the 2 or the 7? (the 7) Explain that the 2 is called the tens digit and the 7 is called the ones digit. Ask students to tell you the tens digit and the ones digit in various numbers.

Exercises: Write the ones digit.

a) 54 ones digit: b) 68 ones digit:

c) 38 ones digit: d) 76 ones digit:

e) 74 ones digit: f ) 91 ones digit:

Bonus: 8 ones digit:

Answers: a) 4, b) 8, c) 8, d) 6, e) 4, f ) 1, Bonus: 8

Write “7” on the board and tell students you want to represent the number using tens and ones blocks. ASK: How many ones blocks will I need? (7) Will I need any tens blocks? (no) Show the number 7 using seven ones blocks and no tens blocks. ASK: How many ones blocks did I use? (7) What is the ones digit of the number 7? (7) How many tens blocks did I use? (0) What is the tens digit in the number 7? (0) Explain to students that they can write the number 7 as “07” (write “07” on the board) to show that the tens digit is 0, but usually people leave out the 0 and write just “7.” Repeat with the number 5.

Exercises: Write the tens digit.

a) 54 tens digit: b) 68 tens digit:

c) 38 tens digit: d) 76 tens digit:

e) 14 tens digit: f ) 91 tens digit:

Bonus

g) 9 tens digit: h) 03 tens digit:

Answers: a) 5, b) 6, c) 3, d) 7, e) 1, f ) 9, Bonus: g) 0, h) 0



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Then write 25, 34, and 35 on the board and ask a volunteer to circle the two numbers with the same tens digit and underline the two numbers with the same ones digit. Repeat with similar sets of numbers.

Exercises: Circle the two numbers with the same tens digit. Underline the two numbers with the same ones digit.

a) 74 84 89 b) 51 58 18 c) 45 54 44 d) 61 71 75

Bonus: 51 12 35 48 50 84 25

Answers: a) 74 84 89, b) 51 58 18, c) 45 54 44, d) 61 75 71,

Bonus: 51 12 35 48 50 84 25

ACTIVITIES 2–3 (Essential)

2. I Have —, Who Has —? (see unit introduction) Use BLM Game Cards to make cards with numerals and base ten models. EXAMPLE: 38 on the top and 2 tens blocks with 5 ones blocks on the bottom (I have 38, who has 25?).

3. Have students represent the numbers using tens and ones blocks, and then have them draw a model of the tens and ones blocks using grid paper or BLM 1 cm Grid Paper.

a) 23 b) 37 c) 22 d) 45

Sample answer: a)

Extensions1. Have students stack as many ones blocks as they can in a given time

interval. Then ASK: Did you stack more than 10 or less than 10? How can you tell? (compare to a tens block) More than 20 or less than 20? (compare to two tens blocks) Have students determine how many ones blocks they stacked by counting the number of tens blocks and then the number of extra ones blocks they need to build an equivalent stack. Repeat several times.

2. Write “22” on the board. Explain that there are two 2s in the number, but they mean different things. Point to the first 2 and ASK: What does this 2 mean? (2 tens) Point to the second 2 and ASK: What does this 2 mean? (2 ones) Emphasize that the tens digit in any number always tells how many tens, and the ones digit tells how many ones. Repeat with more two-digit numbers that have the same tens digit as ones digit, such as 55, 77, and 99.


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3. Write “45” and “54” on the board. ASK: What does the number 5 mean in 45? (5 ones, or 5) What does the number 5 mean in 54? (5 tens, or 50) Emphasize that what the 5 means depends on the position of the 5. If 5 is the tens digit, it means 5 tens; if 5 is the ones digit, it means 5 ones. Repeat with 57 and 74, asking what the 7 means in each number. Then have students write what the 9 means in each number below.

a) 19 b) 94 c) 39 d) 9 e) 90 f ) 09

Answers: a) 9 ones, or 9; b) 9 tens, or 90; c) 9 ones, or 9; d) 9 ones, or 9; e) 9 tens, or 90; f ) 9 ones, or 9



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NS2-21 Ordering Numbers to 100Pages 108–113

GoalsStudents will compare and order numbers to 100.


Can find numbers in a hundreds chart

MATERIALS

red and blue tens and ones blocksBLM 1 cm Grid Paper (p M-7)metre sticks or tape measures to 100 cmBLM Hundreds Chart (p M-5)tokens or counters

Use tens and ones blocks to compare numbers to 100. Write “29” on the board in blue and “34” in red. Give students more than enough tens and ones blocks to make 29 using blue and 34 using red. Remind students that 1 tens block is the same as 10 ones blocks. Have students select red and blue tens and ones blocks to represent the two numbers. Then have students place the blue blocks beside the red blocks. Emphasize that students must place tens blocks with tens blocks and ones blocks with ones blocks. ASK: Which number is larger, 29 or 34? How do you know? (There is 1 red tens block left over and 5 blue ones blocks left over. The number with more left over is larger. Ten is more than 5, so 34 is larger than 29.) Repeat with 42 (blue) and 37 (red). ASK: Do you have more red or blue left over? (blue) Which number is more? (42) Repeat with various numbers. Have students use the same method for the exercises below.

Exercises: Which number is larger?

a) 16 or 21 b) 83 or 38 c) 38 or 74

Answers: a) 21, b) 83, c) 74

Use tens and ones blocks symbolically to compare two numbers. Write two numbers on the board, say 36 and 43. Ask students to predict which number is bigger. Draw the base ten representation for the majority vote on the board. Then have a volunteer try to make the other number by colouring the picture on the board. For example, if students predict that 43 is bigger, the volunteer will colour 3 tens blocks and 6 ones blocks for the smaller number.

If students predict that 36 is bigger, the volunteer will not be able to colour 4 tens blocks and 3 ones blocks; there won’t be enough blocks drawn on the board. Repeat with various numbers. Have students use this same method for the following exercises, using grid paper or BLM 1 cm Grid Paper.


VOCABULARYhundreds chartlargerlargestsmallersmallest


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Exercises: Which number is larger?

a) 22 or 17 b) 65 or 56 c) 49 or 71

Answers: a) 22, b) 65, c) 71

Use a number line to order numbers. Give students a long number line to 100 (such as a metre stick, a tape measure, or a number line made from a hundreds chart as described in the unit introduction) and challenge them to find these numbers: 38, 12, 25. ASK: Which is the smallest number? (12) Which is the largest number? (38) How do you know? (the numbers farther along on the number line are larger) How do we write the numbers in order from smallest to largest? (12, 25, 38) How do we write the numbers from largest to smallest? (38, 25, 12) Repeat with more groups of three numbers.

Exercises

1. Write the numbers from smallest to largest.

a) 28, 24, 31 b) 41, 39, 40 c) 78, 29, 56

Bonus: 42, 14, 74, 41, 32, 73

Answers: a) 24, 28, 31; b) 39, 40, 41; c) 29, 56, 78; Bonus: 14, 32, 41, 42, 73, 74

2. Write the numbers from largest to smallest.

a) 47, 53, 19 b) 59, 61, 58 c) 87, 68, 79

Bonus: 3, 57, 32, 25, 46, 91


Discuss how ordering numbers using a number line is harder or easier than using blocks. (possible answers: it is harder to find numbers on the number line than to make them using blocks, but once we find the numbers, it is easier to compare them; it is much easier to compare many numbers on a number line)

Use a hundreds chart to order numbers. Review finding numbers on a hundreds chart. Students can compare many numbers at a time by first finding each number on the chart and then writing them in order. Give each student a copy of BLM Hundreds Chart for the exercises below. Students can place tokens or counters on the numbers in the chart and then write the numbers in order on a separate sheet of paper.

Exercises: Write the numbers from smallest to largest. Use a hundreds chart.

a) 34, 21, 26, 19, 7, 45 b) 21, 12, 33, 9, 41, 14

c) 31, 62, 77, 80, 43, 52

Answers: a) 7, 19, 21, 26, 34, 45; b) 9, 12, 14, 21, 33, 41; c) 31, 43, 52, 62, 77, 80



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ACTIVITY (Optional)

Find page numbers. Have students open their JUMP Math AP books to page 1. Then have them turn and point to the page numbers in the following order: 24, 29, 26, 21, 28, 20, 25, 27, 30, 34, 31, 38, 36, 39, 35, 37, 41, 48, 96, 45, 49, 82, 47, 44. Ensure students are turning the pages in the correct direction.

Compare numbers with the same number of tens. Tell each student to take 3 red ones blocks and 5 blue ones blocks. ASK: Which is more, 3 or 5? Have a volunteer show the answer by matching the blocks. Repeatedly have students add a tens block to each group. After each addition, ask students what numbers they have and which number is bigger. Emphasize that by adding a tens block to each, we never change which number is bigger. When two numbers have the same number of tens, the number with more ones is bigger.

Exercises: Write the numbers from largest to smallest.

a) 42, 48, 45 b) 56, 52, 59 c) 98, 90, 91

Bonus: 59, 52, 58, 51, 56, 55


Compare numbers with different numbers of tens. Tell each student to take 3 red ones blocks and 6 blue ones blocks. ASK: Which is more? How do you know? Tell each student to add 2 red tens blocks and 1 blue tens block. Repeat the questions. This time, the colour of the larger number changed: even though there are more blue ones blocks, there are more red blocks altogether. ASK: How can a number with 3 ones blocks be more than a number with 6 ones blocks? (it has more tens blocks) Repeat with more numbers such as: 31 and 26; 37 and 45. Emphasize that the number with more tens is bigger.

Exercises

1. Circle the larger number.

a) 52 49 b) 37 51 c) 60 8

d) 77 81 e) 9 61 f ) 92 29

Answers: a) 52, b) 51, c) 60, d) 81, e) 61, f ) 92

2. Write the numbers from smallest to largest.

a) 24, 59, 6, 28, 61, 52 b) 79, 91, 75, 49, 80, 90

c) 51, 2, 27, 36, 65, 50

Answers: a) 6, 24, 28, 52, 59, 61; b) 49, 75, 79, 80, 90, 91; c) 2, 27, 36, 50, 51, 65


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Extensions1. Give each student 9 tens blocks and 9 ones blocks. Challenge students

to find as many answers as they can for the following problems.

a) Find a number between 42 and 82 that can be made with exactly 9 blocks.

b) Find a number between 12 and 45 that can be made with exactly 7 blocks.

c) Find a number between 73 and 91 that can be made with exactly 8 blocks.

d) Find a number between 73 and 91 that can be made with exactly 11 blocks.

Answers: a) 45, 54, 63, 72, 81; b) 16, 25, 34, 43; c) 80, d) 74, 83

2. Provide each student with 10 ten-frames, or show students how to cut their own ten-frames from grid paper (they can use BLM 1 cm Grid Paper). Have students write the numbers below in order from smallest to largest, and verify their answers using ten-frames. Students can use ones blocks to fill part of a ten-frame.

a) 45, 32, 49 b) 52, 31, 49

c) 87, 78, 76 d) 91, 89, 90

Answers: a) 32, 45, 49; b) 31, 49, 52; c) 76, 78, 87; d) 89, 90, 91

3. Marko ordered the numbers from smallest to largest. He made some mistakes. Find Marko’s mistakes and explain them.

a) 43, 45, 54, 50, 61 b) 73, 81, 28, 93, 99

c) 29, 35, 72, 81, 79 Bonus: 29, 37, 84, 91, 100, 99

Answers: a) 54 and 50 are in the wrong order; 54 is larger than 50 since they have the same number of tens and 54 has more ones; b) 28 should be the first number; 28 has fewer tens than all the other numbers; c) 81 and 79 are in the wrong order; 79 is smaller than 81 since it has fewer tens; Bonus: 100 should be the last number since it is larger than all the other numbers (it has 10 tens)



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NS2-22 Adding, Subtracting, and OrderPages 114–116

GoalsStudents will see that changing the order of the numbers in addition

does not change the total; however, changing the order of numbers in subtraction sentences does change the situation.


Knows the plus (+) and minus (–) signsCan add and subtract within 20Can distinguish between right and left

MATERIALS

two-colour counters or blocks of two different colourslarge blank paper dominodominoes

Switch objects between hands to show that the total stays the same. Make sure students can correctly identify their left and right hands. Hold 3 objects in your left hand and 4 in your right hand. ASK: How many do I have in each hand? How many do I have in total? Write on the board:

+ =

left hand right hand total

Have a volunteer fill in the correct numbers. Then switch hands and have the volunteer write the new corresponding addition sentence (left hand first again). ASK: What is the same about the two addition sentences? (the numbers that are added together and the totals) What is different? (the order of the numbers added together) Repeat with several examples.


1. Have students work in pairs, facing each other. Partner 1 holds up some fingers on their left hand (for example, 4), and some fingers on their right hand (for example, 3). Reading from left to right, one partner sees 4 + 3, the other sees 3 + 4. Students record the two sentences and the totals. Students alternate roles and repeat several times.

ASK: Does the order you add the numbers change the total? (no)

Add three numbers and switch the order. Students work in pairs. One partner picks up some counters with one hand and some with the other hand. The other partner picks up some counters only with the non-writing hand. The person with a free hand records different number sentences


VOCABULARYaddition sentencesubtraction sentencetake away fromtotal


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to show the total number in all three hands by counting different hands first. Challenge students to find at least three different number sentences. NOTE: If two of the numbers are the same (e.g., 2, 3, 3), there will only be three number sentences. If all numbers are the same, there will only be one number sentence. If the three numbers are different, there will be a total of six number sentences.

Turn dominoes around and add. Tape a blank paper domino to the board. Have a volunteer draw dots on the domino to show 6 + 4. ASK: What could I do to this domino to make it show 4 + 6 instead of 6 + 4? (turn it around) Does turning the domino change the total number of dots? (no) How does turning the domino change the addition sentence? (6 + 4 = 10 becomes 4 + 6 = 10) What stays the same? (the three numbers used and the total) What is different? (the order of the other two numbers) Give students dominoes. Have them turn the dominoes around to write two addition sentences.


2. Toss 8 two-colour counters. ASK: How could the colours show an addition sentence? Could we count red first and then yellow? What number sentence would we get? What if we counted yellow first and then red—what number sentence would we get?

Give each pair of students up to 10 two-colour counters. Partner 1 tosses the counters and Partner 2 writes two addition sentences (yellow + red = total, red + yellow = total). Students switch roles and repeat several times.

Order doesn’t matter in addition. Students have seen that the order of numbers does not matter in addition. As a reminder, write on the board: 6 + 2 = 2 + 6. Emphasize that we always read from left to right, so 6 + 2 means start with 6 and add 2, and 2 + 6 means start with 2 and add 6. Verify that these are equal with a picture (e.g., draw 6 circles and then add 2 more, then draw 2 circles and add 6 more, for a total of 8 both times).

Does order matter in subtraction? Write on the board: 6 − 2 = 2 − 6. SAY: 6 – 2 means start with 6 and take away 2. What does 2 − 6 mean? Emphasize that we start from the left, so we have to start with 2 things and try to take away 6 of them. Show 2 objects. Ask a volunteer to take away 6 of them. Explain that the question doesn’t make sense: if you start with 2 things, you can’t take away 6 of them. Give students counters and have them decide whether they can subtract the following: 3 − 4 and 4 − 3. Have volunteers show and explain their answers using the counters. (Sample answer: 4 is more than 3, so you cannot subtract 4 from 3. So 3 − 4 doesn’t make sense. Since 3 is less than 4, 4 − 3 makes sense; you can take 3 away from 4. 4 − 3 = 1)

Draw on the board:



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ASK: How many circles are on the board? (4) Can you take 6 away from 4? (no) Why not? (because 6 is more than 4) Ask if there is any student who thinks they can take 6 away from 4. If there is no volunteer, say that you will try. Start crossing out circles and counting “1, 2, 3, 4” as you cross out each circle:

SAY: Now I’m stuck. I need to take away two more circles, but there are no more circles to take away. Write on the board:

4 − 6 = 6 − 4 =

ASK: Which problem makes sense? (6 − 4) Circle that problem. ASK: Which problem does not make sense? (4 − 6) Cross out that problem. Have a volunteer draw a picture with circles to solve the problem that makes sense. The final picture should look like this:

4 − 6 = 6 − 4 = 2

Exercises: Solve the problem that makes sense.

a) 4 − 7 = or 7 − 4 = b) 5 − 3 = or 3 − 5 =

c) 1 − 9 = or 9 − 1 = d) 10 − 7 = or 7 − 10 =

Answers: a) 7 − 4 = 3, b) 5 − 3 = 2, c) 9 − 1 = 8, d) 10 − 7 = 3

Extensions1. Have students count the number of letters in each sentence by adding

the number of letters in each word:

Jin’s birthday is today. (4 + 8 + 2 + 5 = 19) Today is Jin’s birthday. (5 + 2 + 4 + 8 = 19) Is today Jin’s birthday? (2 + 5 + 4 + 8 = 19)

ASK: What do you notice? (the total is always 19, the same 4 numbers are in all the sentences) Do you know why the answer is always the same? (the same 4 words are in all the sentences, just rearranged)

2. Solve the problem that makes sense. Use base ten blocks to help you.

a) 25 − 10 = or 10 − 25 =

b) 45 − 20 = or 20 − 45 =

c) 20 − 34 = or 34 − 20 =

d) 49 − 50 = or 50 − 49 =

Answers: a) 25 − 10 = 15, b) 45 − 20 = 25, c) 34 − 20 = 14, d) 50 − 49 = 1

More sentences: The blue hat is bigThe big hat is blue.Is the big hat blue?Is the blue hat big?


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3. Solve the problem that makes sense. Write the answer as a number word.

a) five minus three or three minus five

b) nine minus eleven or eleven minus nine

c) eighteen minus twelve or twelve mine eighteen

d) six minus twenty or twenty minus six

Answers: a) five minus three equals two, b) eleven minus nine equals two, c) eighteen minus twelve equals six, d) twenty minus six equals fourteen

4. Provide each student with a copy of BLM Subtract Two Ways (p H-49). Work through the first example together, which shows how to solve 9 − 2 − 4, and 9 − 4 − 2. ASK: What is the same about these two problems? (the three numbers are the same: 9, 4, and 2; also, the answer is the same: 3) What is different? (the order of 4 and 2) When you are subtracting two smaller numbers (like 2 and 4) from a larger number (like 9), do you think the order of the two smaller numbers will matter? Accept yes or no answers at this point. After students complete the BLM, discuss this question again. Remind students that when subtracting two numbers (a smaller number from a larger number), the order of the numbers matters: the larger number must be written first otherwise the problem doesn’t make sense. However, when subtracting two smaller numbers from a larger number, as long as the larger number is written first, the order of the two smaller numbers doesn’t matter.



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NS2-23 Adding with a Number LinePages 117–120

GoalsStudents will locate numbers on a partial number line and then use

number lines to add.


Can count to 100Can order numbersUnderstands the concept of moreCan add by counting on

MATERIALS

BLM Frog (p H-50)BLM Blank Number Lines (p H-51)BLM Finding Missing Numbers (p H-52)

Locating numbers on a number line. Draw a partial number line from 34 to 41:

34 35 36 37 38 39 40 41

Tell students that you want to find 39. Start at 34 and ASK: Is this it? (no) Try 35, 36, and 37. SAY: I wonder if we missed 39. I’m at 37 now and we haven’t found it yet. How can we be sure we didn’t miss it? (39 is greater than 37) PROMPT: What comes first, 37 or 39? Emphasize that as long as you’re still at numbers that come before 39, then you know you didn’t miss it. Continue searching one at a time for 39 until you find it. Repeat with other numbers on partial number lines.

Introduce the strategy of starting in the middle. Draw a number line from 46 to 56 and tell students you want to find 53. Explain that instead of starting at 46 and checking all the numbers until you get to 53, you’re going to take a shortcut. Start in the middle of the number line (at 51) and decide whether to go right or left. ASK: Is 53 more than 51 or less? (more) Which way should I go on the number line: right or left (this way or that way)? Look to the right of 51 because 53 is more than 51. Explain that now you have fewer numbers to check. Repeat for various number lines and numbers.

Adding 1 on a number line. Remind students that to find 3 + 1, they can find the number they say after 3—it is the number that is one more than 3. Draw a number line on the board and tell students that instead of counting on from 3 and saying the next number, they can draw a leap from 3 to the next number. Cut the frog out of BLM Frog. Place the frog on the 3 and move it one leap forward to the 4.0 1 2 3 4 5 6 7


VOCABULARYnumber lineleap


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The frog ends up at the next number after 3, or 3 + 1. Write: 3 + 1 = 4. Add 1 to several other numbers. Use partial number lines that begin at larger numbers. EXAMPLE: On a number line from 34 to 41, find 37 + 1. At first, place the frog on the number line where students need to start and have them just add 1. Then have students both locate the starting number and add 1. Students can draw a dot (instead of a frog) at the starting number and draw arrows for the leaps. (as shown in the margin)

For the following exercises, provide each student with BLM Blank Number Lines. Write the numbers 0 to 10 on each number line before photocopying, or have students write these numbers. Students draw a dot for the starting number and arrows for leaps.

Exercises: Use the number line to add 1.

a) 3 + 1 = b) 7 + 1 = c) 5 + 1 = d) 9 + 1 =

Answers: a) 4, b) 8, c) 6, d) 10

Adding 2 or 3 on a number line. Start by drawing two leaps in order to add 2. EXAMPLES: 4 + 2, 27 + 2, 72 + 2, 38 + 2, 46 + 2. Then draw three leaps in order to add 3. EXAMPLES: 5 + 3, 17 + 3, 26 + 3, 39 + 3. At first, draw a big dot where students need to start, then have students do both steps (finding the place to start and drawing the leaps).

Then mix up examples that require adding 1, 2, or 3. Students need to decide how many leaps to add—1, 2, or 3—depending on the second addend.

Connect adding on a number line to adding by counting on. Draw a number line from 0 to 10 on the board and tell students that you want to add 5 + 3. Have a volunteer demonstrate by counting on starting from 5 using their fingers. Then tell students you will add 5 + 3 using the number line. ASK: What number should I start at? (5) Draw a big dot at the 5. ASK: What part of counting on to add is this like? (saying 5 with your fist closed) How many leaps should I draw starting at the 5? (3) What part of counting on is this like? (saying the next three numbers after 5)

Connect the number sentence to the number line. Write the addition sentence (5 + 3 = 8) below the number line. Point out that leaps start and end at numbers that we see in the addition sentence. Show that 5 is where the leaps start and 8 is where they end. Connect the numbers in the number sentence to the numbers on the number line.

4 5 6 7 8 9

5 + 3 = 8

34 35 36 37 38 39 40 41



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Repeat with several examples, having volunteers join the numbers. Then draw number lines and have students fill in the missing numbers in the number sentences based on the number lines.

4 5 6 7 8

+ 2 =

Find the missing number. Draw more number lines that show addition but leave out the second number in the addition sentence. Include examples that require more than three leaps.

Have students practise writing addition sentences for models and vice versa. Copy BLM Blank Number Lines onto overheads, mark the starting point, draw the leaps, and have students write the corresponding addition sentences. Or write number statements on the board (e.g., 24 + 5) and have students draw the corresponding models and solve the addition on laminated (re-usable) copies of the BLM.

Extensions1. Have students compare adding 3 + 4 and 4 + 3 on the same number

line. They can do 3 + 4 above the line and 4 + 3 below it. Give several examples of this sort. ASK: What do you notice?

2. Have students draw their own number lines to show and solve the addition. Students can use grid paper, or you can provide them with BLM Blank Number Lines.

a) Draw a number line from 10 to 20 and show 13 + 6 = .

b) Draw a number line from 30 to 40 and show 44 + 3 = .

c) Draw a number line from 60 to 70 and show 62 + 5 = .

d) Draw a number line from 80 to 90 and show 81 + 8 = .

Answers: a) 19, b) 47, c) 67, d) 89

3. Have students complete BLM Word Problems (p H-53).

Answers: 5 + 2, 7; 4 + 5, 9

EXAMPLES: 43 + = 4851 + = 57 58 + = 63

EXTRA PRACTICE

BLM Finding Missing Numbers


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NS2-24 Adding by Counting OnPages 121–124

GoalsStudents will review adding by counting on, and will discover that adding

by counting on is easier when starting from the larger number.


Can order numbers to 100Can count to 100 orally and in writingCan addKnows that numbers can be added in any order

MATERIALS

ball or paper objecta hundreds chart for displaypaper domino modelling 2 + 7

Add 1 by finding the next number. Draw three circles on the board. Count the circles one at a time and write the numbers above the circles as you count, as shown below:

1 2 3

Then add one more circle and ASK: Now how many circles are there? Erase the numbers above the circles and count again. Rewrite 1, 2, and 3, and add 4 above the last circle. Repeat for eight circles and then nine circles.

Draw five circles on the board and count them, writing the numbers above the circles as you count. Then, instead of erasing the original counting, SAY: I might as well leave the numbers there and just write the next number above the new circle. Add another circle and ASK: What is the next number after 5? (6) Write “6” over the last circle, and then write the addition sentence. The final picture should look like this:

1 2 3 4 5 6

5 + 1 = 6

Repeat with several examples where students add 1 to a number, but have volunteers finish the model. Emphasize that the answer is just the next number you say when counting. Write on the board the sequence of numbers from 0 to 10. Emphasize that the numbers are written in order and have students add 1 to more one-digit numbers without drawing pictures. EXAMPLES: 4 + 1, 8 + 1, 0 + 1. Continue writing the numbers up to 20 on the board to help students complete the following exercises.


VOCABULARYnext



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Exercises: Add.

a) 15 + 1 = b) 18 + 1 =

c) 12 + 1 = d) 17 + 1 =

Bonus

e) 73 + 1 = f ) 52 + 1 =

g) 65 + 1 = h) 39 + 1 =

Answers: a) 16, b) 19, c) 13, d) 18, Bonus: e) 74, f ) 53, g) 66, h) 40

Add 2 by finding the next two numbers. Draw the picture in the margin on the board. Repeat the process for adding 1, but add 2, first to numbers from 0 to 10, then to numbers up to 20. EXAMPLES: 4 + 2, 8 + 2, 13 + 2, 16 + 2.

ACTIVITY (Essential)

Catch. (see unit introduction) Students say the next number after the one you say. Repeat with students saying the next two numbers. Finally, students just say the number plus 2.

Review counting on your fingers. Have students count to 10 on their fingers, starting with the thumb. Then hold up several fingers at once as though you counted this way and ask students what number you counted to.

Use your fingers to keep track of how many numbers you said. SAY: I would like to add 6 + 8, but saying 8 numbers after 6 is a lot of numbers to keep track of. I am going to use my fingers to help keep track. I can hold up one finger for every number I say after 6. Demonstrate and ASK: What is the first number that comes after 6? (hold up your thumb when students say 7) And the next number? (hold up your thumb and forefinger when they say 8) Continue in this way. After 10, ASK: How many numbers have I said after 6 so far? (4) How do you know? (you’re holding up 4 fingers) How will I know when to stop? (8 fingers will be up) Continue to count. When you have 8 fingers up, ASK: What was the last number we said? (14) Write on the board: 6 + 8 = 14. Verify by drawing 6 coloured circles and 8 blank circles all in a row. Write 6 on top of the last coloured circle, then write the next eight numbers on top of the blank circles. Count the blank circles to show eight numbers are counted after you said 6.

Exercises: Add by counting on. Use your fingers.

a) 4 + 5 b) 13 + 6 c) 27 + 8 d) 56 + 9

Bonus: 89 + 10

Answers: a) 9, b) 19, c) 35, d) 65, Bonus: 99

1 2 3


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Mistakes in counting on. Model mistakes in counting on to add. EXAMPLES: count faster or slower than you hold fingers up (i.e., only hold up 2 fingers while adding 3); skip or repeat numbers; say some numbers in the wrong order. Challenge students to say what you do wrong each time.

Compare adding by counting on to adding with a hundreds chart. Write on the board: 5 + 3. Colour only square 5 on a hundreds chart. SAY: Starting at 5 is like saying 5 with your fist closed. Then circle the next 3 squares and SAY: Circling the next numbers is like saying 6, 7, 8 while raising 3 fingers one at a time. Instead of seeing how many numbers you have circled, you see how many fingers you have up.

Order in addition doesn’t matter (review). Tape a paper domino showing 2 and 7 to the board. SAY: I want to know how many dots there are.

Cover the 2 and SAY: I know there are 2 dots here, so I can count on, starting at 3. Demonstrate doing so while pointing to each of the 7 dots on the right side: 3, 4, 5, 6, 7, 8, 9. Write: 2 + 7 = 9. Now turn the domino around. Cover the 7 on the left side and SAY: I know there are 7 dots here, so I can count on starting at 8. Demonstrate doing so while pointing to each of the 2 dots on the right side: 8, 9. Write: 7 + 2 = 9. ASK: I added 2 + 7 by counting on and I added 7 + 2 by counting on—did I get the same answer both ways? (yes) Why did that happen? (because order doesn’t matter in addition)

Choosing which number to count on from. Write on the board:

2 + 9 = 9 + 2 =

ASK: Will these questions have the same answer? (yes) How do you know? (because order doesn’t matter in addition) Explain that since you know that both problems have the same answer, you want to do the easier one. SAY: Let’s try it both ways and find out which one is easier. ASK: How would I solve 2 + 9? How many numbers would I count? What number would I start at? (count 9 numbers starting at the number after 2) Demonstrate doing so. How would I solve 9 + 2? (count 2 numbers starting at the number after 9) Demonstrate doing so. ASK: What is easier—to count 9 numbers starting at 3 or to count 2 numbers starting at 10? (Demonstrate both again.) Which would be faster? (counting 2 numbers starting at the number after 9) Emphasize that when mathematicians see two problems that they know have the same answer, they can be smart and pick the easier one to do.

Have students predict which will be faster to do: 3 + 7 or 7 + 3. Have students try it both ways and have volunteers circle the faster and easier way. Repeat with the examples in the margin. Tell students to look at the questions they circled as being easier. Point to the first number of each circled question and ASK: Is this number the bigger number or the smaller number? What do students notice? (the first number is always the bigger number) Explain to students that they can make 4 + 9 into an easier problem just by writing the bigger number first: 9 + 4. Demonstrate adding by counting on both ways.

3 + 7 7 + 3

8 + 4 4 + 8

1 + 9 9 + 1

2 + 10 10 + 2

5 + 1 1 + 5

8 + 3 3 + 8



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Writing blanks to add by counting on. Challenge volunteers to try adding more numbers both ways using blanks. At first, give volunteers the numbers to count from and the number of blanks. EXAMPLE:

6 + 3 6

3 + 6 3

Then have volunteers write the numbers and blanks, and discuss how they know how many blanks to draw (start at the first number and draw the second number of blanks or vice versa). Have students use the same method for the following exercises.

Exercises: Draw the correct number of blanks. Add by counting on.

a) 3 + 8 b) 9 + 1 c) 10 + 2 8 + 3 1 + 9 2 + 10

Bonus: 23 + 4 and 4 + 23

Sample solution: a) 3 + 8: 3 4 5 6 7 8 9 10 11 (3 + 8 = 11); 8 + 3: 8 9 10 11 (8 + 3 = 11)

ASK: Which is easier—to count starting from the bigger number or the smaller number? (from the bigger number) Why do you think that is? (there are fewer numbers to count)

Extensions1. Tell students you are going to play a word game where some letters

are worth more points than others, and the points are written right next to the letter. SAY: I have the following letters:

A1 B3 I1 T1 S1 H4 Y4

Have students determine how many points various words are worth by counting on:

HI (4 + 1 = 5)

IS (1 + 1 = 2)

BY (3 + 4 = 7)

Bonus: SHY, BAY, SAY, THIS

Students who finish quickly can make their own words using the same letters and count the total points for each word.

Then show these letters: A1 C3 E1 H4 O1 B3 F4

How many points is each of these words worth? CAT, BET, HAT, THE

Bonus: BATH, CHAT, BOAT, COAT, TEACH

Again, students who finish quickly can make their own words from the letters and count the total points for each word.


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2. Show students how to add 11 by counting on using their fingers. After they reach 10 and run out of fingers, they just need to say one more. Repeat with 12, where they have to say two more. Then have them complete the following exercises by counting on in this manner.

a) 15 + 11 = b) 19 + 11 = c) 23 + 11 =

d) 30 + 12 = e) 45 + 12 = f ) 68 + 12 =

Bonus: 80 + 13 =

Answers: a) 26, b) 30, c) 34, d) 42, e) 57, f ) 80, Bonus: 93

3. Have students complete BLM Missing Addends (p H-54). Emphasize that they should use counting on to figure out how many more dots they need on the dominoes.

Answers: 3 dots, 3; 6 dots, 6; 3 dots, 3; 1 dot, 1; 6 dots, 6; 3 dots, 3; 4 dots, 4; 4 dots, 4; 7 dots, 7



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NS2-25 Subtracting with a Number LinePages 125–129

GoalsStudents will subtract by drawing leaps on a number line.


Can count to 10Understands the concept of moreUnderstands arrows and directionUnderstands addition on a number lineKnows to say the number before to subtract 1Knows to say the number before two times to subtract 2

MATERIALS

masking tape (optional)BLM Blank Number Lines (p H-51)BLM Which Way Does the Frog Leap? (p H-55)

Draw circles in a row to subtract. Draw 7 circles in a row (because circles are easy to draw) and count them by writing the numbers above the circles. Cross out the fourth circle and then count the remaining (leftover) circles by writing the numbers underneath the circles:

1 2 3 4 5 6 7

1 2 3 4 5 6

Write the subtraction sentence: 7 − 1 = 6.

Taking away the last circle(s) makes it easier to count the leftover circles. Draw several rows of 7 circles on the board with a different circle crossed out in each row:

1 2 3 4 5 6 7

1 2 3 4 5 6 7

1 2 3 4 5 6 7

Have volunteers count the remaining circles by writing the numbers underneath. ASK: Did we always get the same answer? (yes, 6) Where are the numbers under the circles the same as the numbers above the circles? (before the crossed out circle) Why did that happen? (once a circle is crossed out, the count changes) How is counting the leftover circles easier when you take away the last circle? (just look at the numbers above the circles; the numbers underneath will be the same as the numbers above)

Have students count how many circles are left over:

1 2 3 4 5 6 7 8


VOCABULARYbackwardsforwardsleftoverminusnumber lineremainingsubtracttake away


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ASK: How did you count the leftover circles? Emphasize that when you take away 1 circle, you just have to move back 1 number to find the number of leftover circles.

Exercises: Draw circles and number them. Cross out the last circle to subtract. Subtract 1.

a) 5 − 1 = b) 3 − 1 = c) 9 − 1 =

Selected answer: a) 1 2 3 4 5; 5 − 1 = 4

Subtract 1 using a number line. Draw a number line from 0 to 8. Discuss how you can subtract 1 using the number line by drawing a leap going backwards from the 8. SAY: We don’t need to draw circles; we can just use the number line to find the number before. (You may wish to create a large number line on the floor with masking tape so students can act out the leap.)

0 1 2 3 4 5 6 7 8

Show students several number lines with a leap drawn backwards and ask them to solve the subtraction problems. EXAMPLES:

0 1 2 3 4 5

5 − 1 =

40 41 42 43

43 − 1 =

60 61 62 63 65 65 66

66 − 1 =

Subtracting larger numbers. Repeat the above for subtracting 1 but use 2 leaps to subtract 2. Then subtract larger numbers. Ensure students understand that the number of leaps is the number being subtracted. Use the same process for subtracting on a number line as for adding:

•Useabigdottoshowwheretostartsubtractingfrom(findthefirstnumber on the number line).

•Decidehowmanyleapstodrawbylookingatthesubtractionsentence (the number being subtracted tells how many leaps to draw).

•Decidehowtofindtheanswertoasubtractionsentencefromanumber line (where the leaps end).

Provide students with a copy of BLM Blank Number Lines for the following exercises. You might write the numbers 0 to 10 on each number line before photocopying, or have students write the numbers on their copies.



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Exercises: Use a number line. Draw a dot on the starting number. Draw leaps back for the second number. Subtract.

a) 7 − 3 = b) 10 − 4 =

c) 5 − 2 = d) 9 − 6 =

Answers: a) 4, b) 6, c) 3, d) 3


1. Have students form a human number line (from 1 to however many students are in the class). Each student has a number, which they need to remember; ensure that everyone knows their number by asking in random order. Give the student with the highest number a ball. This student says a subtraction sentence starting with his/her number and students use the ball to count off “leaps” on the line to solve the subtraction sentence. EXAMPLE: Player 17 might say 17 − 4. Player 17 then tosses the ball to Player 16 and the class counts 1, 16 tosses to 15 and the class counts 2, and so on to 4. When the class counts 4, the player who caught the ball says “17− 4 is 13” (that’s the player’s number). Player 13 then says a new subtraction sentence and play continues until the ball reaches player 1.

Choosing between adding and subtracting. Tell students that you want to find 7 − 3. ASK: How can I use a number line to help me? Take answers and then draw a number line on the board. Invite a volunteer to put a big dot where you should start drawing leaps. Then ASK: How many leaps should I draw on the number line? (3) Should I go forwards or backwards? (backwards) How do you know? What symbol shows us that the leaps go backwards instead of forwards? (the minus sign) Repeat with more addition and subtraction problems. Then give students BLM Which Way Does the Frog Leap?

Provide students with another copy of BLM Blank Number Lines for the following exercises.

Exercises: Write the first number near the middle of the number line. Finish the number line. Use the number line to add or subtract.

a) 37 − 3 = b) 22 + 4 =

c) 44 − 4 = d) 58 + 3 =

Answers: a) 34, b) 26, c) 40, d) 61


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2. Students form a human number line as in Activity 1 and play the same game, but they have the option of adding or subtracting. Start with a middle player. The goal is to make sure everyone plays in as few turns as possible. Students must stay within the limits (e.g., if there are 17 players, Player 15 is not allowed to say 15 + 3).

Extensions1. Have students complete BLM Adding and Subtracting (p H-56).

Students add and subtract at the same time, e.g., 5 − 3 + 2. To help keep track, students might use a different colour to draw the leaps for different numbers. Remind students that we read from left to right, so we do 5 − 3 first because it is the first one we see. Then we add 2 to the answer.

2. Make a subtraction sentence from the group of numbers.

a) 6, 2, 8 b) 7, 8, 1 c) 5, 3, 8 d) 3, 9, 6

Answers: a) 8 − 2 = 6 or 8 − 6 = 2, b) 8 − 7 = 1 or 8 − 1 = 7, c) 8 − 5 = 3 or 8 − 3 = 5, d) 9 − 3 = 6 or 9 − 6 = 3

3. Make a subtraction sentence using three of the four numbers given.

a) 9, 8, 2, 1 b) 5, 3, 4, 2 c) 7, 6, 4, 3

d) 8, 10, 1, 2 e) 4, 3, 7, 10

Answers: a) 9 − 8 = 1 or 9 − 1 = 8, b) 5 − 3 = 2 or 5 − 2 = 3, c) 7 − 4 = 3 or 7 − 3 = 4, d) 10 − 8 = 2 or 10 − 2 = 8, e) 7 − 3 = 4 or 7 − 4 = 3 or 10 − 3 = 7 or 10 − 7 = 3

4. Tasha’s phone number is 633 7523. Her older sister, Nora, remembers their phone number this way: 6 − 3 = 3, 7 − 5 = 2, and 5 − 2 = 3. Can you find subtraction sentences in these phone numbers?

a) 532 8624 b) 871 4312 c) 963 9725

d) 853 9817 e) 880 9909 Bonus: 209 1192

Answers: a) 5 − 3 = 2, 8 − 6 = 2, 6 − 2 = 4; b) 8 − 7 = 1, 4 − 3 = 1, 3 − 1 = 2; c) 9 − 6 = 3, 9 − 7 = 2, 7 − 2 = 5; d) 8 − 5 = 3, 9 − 8 = 1, 8 − 1 = 7; e) 8 − 8 = 0, 9 − 9 = 0, 9 − 0 = 9; Bonus: 20 − 9 = 11, 11 − 9 = 2

5. Working in pairs, each student makes up a subtraction question and draws a number line that includes a range of numbers that helps to solve the problem. Partners exchange papers and solve the subtraction problem using the number line.



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NS2-26 Subtracting by Counting BackwardsPage 130

GoalsStudents will subtract by counting back from 20.


Understands the concepts of more and lessUnderstands that subtraction means “take away”Can identify first and last in a group or listCan count on using fingers

MATERIALS

number cards (optional)

Practise counting back from 5 to 0. Write “0 1 2 3 4 5” on the board and tell students to say the numbers out loud together, but backwards. Point to each number as students say it.

Then erase 2 and leave a space or replace it with a blank line: 0 1 3 4 5. Point to each number and the space as students count backwards from 5, remembering to say 2. Repeat several times with different missing numbers. EXAMPLE: 0 1 2 3 5.

Continue to have students say the numbers in order backwards, but with more missing:

•Twonumbers,butnottwoinarow. EXAMPLE: 0 1 3 5

•Twonumbersinarow. EXAMPLE: 0 1 4 5

•Morethantwonumbers. EXAMPLES: 0 2 5; 3 5

•Allnumbers.

Always point to each number or space starting at the right.

Count back from 10 to 0. Repeat the sequence of questions above, starting from 7, then from 9, and finally from 10.

Count back from any number up to 20. Repeat the sequence of questions above, starting from 13, then 15, then 18, and finally 20.

Counting forwards helps to count backwards. Make the connection between counting forwards and counting backwards. For example, if students are not sure if 8 comes after 9 when counting backwards, they can count forwards from 8 and check to see if 9 comes right after 8.

CURRICULUM REQUIREMENTAB: recommendedBC: recommendedMB: recommendedON: recommended

VOCABULARYbackwardscounting backcounting onminussubtracttake away


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Count backwards between any two given numbers from 0 to 100. Have students count forwards from 40 to 49 and then backwards from 49 to 40. Do the same (count forwards and backwards) for 70 to 79, 60 to 69, and 90 to 99. Then have students count forwards from 38 to 42 and then backwards from 42 to 38. Repeat for 29 to 33 and 67 to 71. Then ASK: What group of numbers comes after the thirties? (forties) What group of numbers comes before the forties? (thirties) What group of numbers comes before the sixties? (fifties) Before the nineties? (eighties) Before the twenties? (teens or 10 to 19) Then have students count back five numbers from: 43, 81, 92, 60, 22.

Exercises: Count back 4 numbers from the given number. Write the 4 numbers.

a) 52 b) 21

c) 79 d) 93

Bonus: Count back 10 numbers.

100

Answers: a) 51, 50, 49, 48; b) 20, 19, 18, 17; c) 78, 77, 76, 75; d) 92, 91, 90, 89; Bonus: 99, 98, 97, 96, 95, 94, 93, 92, 91, 90

Counting back in the real world. Ask students where people use counting back. Examples may include traffic lights (the number of seconds before you have to finish crossing the street), the countdown to the launch of a rocket, the countdown on New Year’s Eve, clocks counting down to the end of a game or match, time left on a microwave oven, and so on.

ACTIVITIES 1–2 (Optional)

1. Prepare a set of cards marked 1 through 10. Shuffle and place the cards face down in two rows of 5. Turn over each card one at a time to find the 10 card. If the card is not a 10, put it back face down in the same spot. If it is a 10, place it face up in a separate discard pile. Then look for the 9 card in the same way and put it face up on the 10 card. Find the 8 card, the 7 card, and so on until the 1 card is placed on the discard pile and there are no cards left. Challenge students to finish as quickly as they can.

Variation: Play with numbers up to 20. (4 rows of 5)

Bonus: Play with numbers up to 50. (10 rows of 5)

2. Pairs put number cards (to 10, 20, or even 100) in a pile face down. Player 1 turns up the top card and Player 2 says the number that comes before. Players then switch roles.

Review subtraction. Remind students how they subtracted 1 from a number by finding the number that comes before and how they subtracted 2 by finding the two numbers that come before.

CONNECTION

Real World



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Subtract any number by counting back. Write the numbers from 1 to 10 on the board. Then write:

7 7 − 3 =

Guide students to fill in the blanks with the next three numbers they would say when counting back from 7 and the answer to the subtraction sentence (which is the last number you say when counting back). Emphasize that in a subtraction sentence, the first number (7) tells us what to subtract from and the second number (3) tells us how many numbers to say when counting back. Have students do more problems individually.

Exercises

1. Count back to subtract.

a) 6 − 2 = 6

b) 8 − 4 = 8

c) 7 − 4 = 7

d) 6 − 3 = 6

Answers: a) 5, 4; 6 − 2 = 4; b) 7, 6, 5, 4; 8 − 4 = 4; c) 6, 5, 4, 3; 7 − 4 = 3; d) 5, 4, 3; 6 − 3 = 3

2. Draw the correct number of blanks. Count back to subtract. Hint: The second number tells you the number of blanks.

a) 10 − 3 = 10 b) 15 − 4 = 15

c) 17 − 5 = 17 d) 16 − 3 = 16

Answers: a) 9, 8, 7; 10 − 3 = 7; b) 14, 13, 12, 11; 15 − 4 = 11; c) 16, 15, 14, 13, 12; 17 − 5 = 12; d) 15, 14, 13; 16 − 3 = 13

3. Write the first number. Draw the correct number of blanks. Count back to subtract.

a) 13 − 5 = b) 18 − 7 =

c) 11 − 7 = d) 16 − 9 =

Bonus: 85 − 10 =

Answers: a) 13: 12, 11, 10, 9, 8; 13 − 5 = 8; b) 18: 17, 16, 15, 14, 13, 12, 11; 18 − 7 = 11; c) 11: 10, 9, 8, 7, 6, 5, 4; 11 − 7 = 4; d) 16: 15, 14, 13, 12, 11, 10, 9, 8, 7; 16 − 9 = 7; Bonus: 85: 84, 83, 82, 81, 80, 79, 78, 77, 76, 75; 85 − 10 = 75

Does it matter which number you count back from? Remind students about adding and order using 6 + 2 as an example. Students can start at 6 and draw 2 blanks or start at 2 and draw 6 blanks:

6 7 8 or 2 3 4 5 6 7 8


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ASK: Does it matter which number you subtract from and how many blanks you draw for 6 − 2? Have students explain their answers. Then draw on the board:

6 5 4 or 2 1 0 ? ? ? ?

Explain that we can’t count back 6 times starting at 2 because there aren’t 6 numbers that are less than 2. We must start with the bigger number (6) and draw the smaller number of blanks (2).

Subtract by counting backwards on your fingers. Write “8 − 2” on the board. SAY: I am going to count back from 8 and I want you to tell me to stop when I have 2 fingers up. Then say “8” with a closed fist (thumb tucked under fingers), “7” with 1 finger (your thumb) up, and “6” with 2 fingers up. If students don’t tell you to stop, ASK: How many fingers am I holding up? Tell students that since you are holding up 2 fingers, you can stop counting, so 8 − 2 = 6. Demonstrate subtracting more numbers from 8 in this way. Repeat until all students actively identify when you have enough fingers up.

Tell students you want them to find 39 − 4. ASK: How many fingers will you be holding up when you get to the answer? (4) Remind students to concentrate on both the numbers they say when counting back and the number of fingers they are holding up. Then ask a volunteer to count back from 39 to find 39 − 4. Repeat with other subtraction sentences starting at 39. Ask how the volunteers knew to start at 39.

Finally, have students subtract from different numbers individually in their notebooks. EXAMPLES: 15 − 3, 48 − 5, 64 − 2.

Potential mistakes in counting back to subtract. Model several incorrect ways of counting back, such as counting faster or slower than you hold fingers up (e.g., hold up 2 fingers while subtracting 3), skipping or repeating numbers, saying some numbers in the wrong order or counting forwards to start (e.g., I get 8 − 4 = 6 because I counted 8 9 8 7 6). Challenge students to tell you what you are doing wrong each time. Then have volunteers subtract by counting back: 7 − 3, 6 − 1, 18 − 3, 49 − 2, 37 − 4, 80 − 5.

Connect subtraction on a number line to subtraction by crossing out circles. Write “6 − 2” and draw a number line from 0 to 6 with 6 circles underneath it. ASK: Which number in the subtraction sentence tells you how many leaps to draw? (the second number) Which number tells you how many circles to take away? (the second number) Emphasize that we draw a leap for every circle we take away. Every time you remove a circle, you move back one number:

0 1 2 3 4 5 6

0 1 2 3 4 5 6

0 1 2 3 4 5 6



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Extensions1. ASK: What pattern is in the answers to these questions: 7 − 1, 8 − 2,

9 − 3, 10 − 4, 11 − 5, 12 − 6? (repeating pattern: 6) Why did that happen? (we kept adding 1 to each of the numbers) Find another set of subtraction problems (at least four problems) that all have the same answer.

2. Have students complete BLM In the Bag (p H-57).

Answers: 9, 8, 10 − 2 = 8; 10, 9, 10 − 1 = 9; 10, 9, 8, 7, 6, 10 − 4 = 6; 10, 9, 8, 7, 6, 5, 4, 10 − 6 = 4

3. Spelling backwards. Challenge students to spell words backwards (to develop the same type of memory skill as counting backwards). EXAMPLES: of, to, in, is, he, on, as, at, the, for, bat, hat, lip, bit, pit, zip, top, mop, zig, and zag. (If they are having trouble, have students spell the words forwards first and then backwards.) When one student has correctly spelled a word backwards, you can ask other students to spell the same word backwards—this will allow more students to participate.

Bonus: Have students spell their own name backwards.

Bonus: Challenge students to think of words that are spelled the same forwards as backwards such as pop, mom, dad, wow, and bib. Give Hints: What do babies wear when they eat? What do you call your mother? And so on.

CONNECTION

Language Arts


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NS2-27 Subtracting by Counting OnPages 131–132

GoalsStudents will use addition to subtract.


Can count back to subtractCan count on to addUnderstands the relationship between addition and subtractionCan find the missing addends in addition problems

Review adding by counting on. Tell students that you want to find 4 + 5 by counting on. SAY: Let’s start at 4 this time. ASK: What number tells you how many fingers to be holding up when you stop? (5) SAY: I will add 4 + 5 using my fingers. Stop me when I have added 5. Say “4” with your fist closed (thumb tucked under), “5” with one finger up (the thumb), and so on until you have five fingers up. If students don’t stop you when you have five fingers up, ASK: How many fingers do I have up? (5) Is that the number that I’m supposed to be holding up when I stop? (yes) What number did I say when I held up my fifth finger? (9) If they don’t remember, SAY: Listen carefully for the number I say when I hold up my fifth finger. That will be the answer to 4 + 5. Then repeat the counting on process. Repeat with more examples.

Finding the missing addend by counting on. Write “4 + = 9” on the board. ASK: How is this problem different from finding the answer to 4 + 5? How can I use counting on to solve the first problem? Explain that instead of knowing that you have to have five fingers up when you stop, you know what number you need to say to stop. ASK: Does anyone know what that number is? (9) Tell students to listen carefully as you count on from 4 and to stop you when you say “9.” Then count on using your fingers and when students tell you to stop, ASK: How many fingers am I holding up? Fill in the blank: 4 + 5 = 9. Repeat with more questions, always asking students to tell you when to stop. Next, have volunteers write the answer on the board after students tell you to stop. Finally, have volunteers count on and write the answer to more missing addend problems.

Subtract by counting on. ASK: How does finding the missing number in 4 + = 9 help you find the answer to 9 − 4 = ? Draw this picture as a reminder:

ASK: How could you use counting on to subtract 9 − 4? If no one answers, SAY: You could count on from 4 and see how many fingers you are holding up when you say “9.” Remind students that what they are really doing is finding the missing number in 4 + = 9, but that’s okay because 9 − 4 has the same answer. Repeat with several examples.


VOCABULARYfewermoresolve



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Compare counting on and counting back to subtract. Ask students if they can think of two different ways of saying “counting on.” (counting up, counting forwards) Ask a volunteer to find 10 − 1 by counting on. Ask another volunteer to find 10 − 1 by counting backwards. ASK: Which way is easier—counting on or counting back? (answers may vary) Which way is faster? (counting back) Why? (When counting back, you only have to say, “10, 9” and see that you are holding up one finger, whereas when counting on from 1, you have to count from 1 to 10 and see that you are holding up nine fingers.) Explain that there are often many ways to solve a math problem, so you can try to pick the best way.

NOTE: In Exercise 3 below, students are guided to discover when counting backwards is a better strategy, and when counting forwards is a better strategy. This will be taught explicitly in the next part of the lesson.

Exercises

1. Subtract by counting backwards, and then by counting forwards.

a) 9 − 8 = b) 9 − 2 = c) 8 − 1 =

d) 8 − 7 = e) 10 − 1 = f ) 10 − 8 =

Selected sample answers: a) counting backwards: 9: 8, 7, 6, 5, 4, 3, 2, 1; 9 − 8 = 1; counting forwards: 8: 9; 9 − 8 = 1; b) counting backwards: 9: 8, 7; 9 − 2 = 7; counting forwards: 2: 3, 4, 5, 6, 7, 8, 9; 9 − 2 = 7

2. For each part in Exercise 1, write which way of subtracting was faster.

Answers: a) counting forwards, b) counting backwards, c) counting backwards, d) counting forwards, e) counting backwards, f ) counting forwards

3. Subtract by counting backwards or counting forwards.

a) 38 − 2 = b) 76 − 73 =

c) 99 − 4 = d) 82 − 79 =

Bonus: Did you use counting forwards or backwards for part a)? Explain your choice.

Answers: a) counting backwards: 38: 37, 36; 38 − 2 = 36; b) counting forwards: 73: 74, 75, 76; 76 − 73 = 3; c) counting backwards: 99: 98, 97, 96, 95; 99 − 4 = 95; d) counting forwards: 79: 80, 81, 82; 82 − 79 = 3; Bonus: Counting backwards, because the second number (2) is very small, so counting backwards is faster.

Understanding when to count forwards or backwards. Write on the board:

98 − 2 =

ASK: Would you count backwards or forwards? (backwards) Lead the students in counting backwards as a class. Then start leading the students in the other method, counting up from 2, but stop after reaching 10.


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ASK: Why was counting back so fast, and why is counting up so slow? (because 2 is a small number; there are only two numbers to count back, but counting up from 2 to 98 would take a long time) If the second number in a subtraction question is small, much smaller than the first number, would you count forwards or backwards? (backwards) Why? (because it would be faster) Write on the board:

98 − 95

ASK: Would you count backwards or forwards? (forwards) Lead the students in counting forwards as a class. Then start leading the students in the other method, counting back 95 numbers from 98, but stop after reaching 90. ASK: Why was counting forwards so fast, and why is counting backwards so slow? (because 95 is a large number, close to 98; counting down 95 numbers would take a long time) ASK: If the second number in a subtraction question is large, close to the first number, would you count forwards or backwards? (forwards) Why? (because it would be faster)

ACTIVITY (Optional)

Missing Number Game. Give each student a strip of paper divided into three equal parts:

Have students write numbers in the first two parts. The first number should have two digits and the second numbers should have one digit.

EXAMPLE: 27 5

Then fold the third part over to cover the second part, so that the second number is hidden, and write the sum of the two numbers on the folded-over flap:

27 27sum

32

Students play with a partner who has to find the missing number. Players can switch roles and then switch partners to play repeatedly.

Students can exchange and solve each other’s problems. Students can check their own work by unfolding the cards. Students can sign the back of each other’s cards when they solve them. You can ask students to get at least 5 signatures on their cards.



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Extensions

1. Complete the subtraction sentence. Write “F” if you counted forwards to subtract and “B” if you counted backwards to subtract.

a) 12 − 11 = b) 12 − 10 =

c) 12 − 9 = d) 12 − 8 =

e) 12 − 7 = f ) 12 − 6 =

g) 12 − 5 = h) 12 − 4 =

i ) 12 − 3 = j ) 12 − 2 =

k) 12 − 1 =

Do you notice any patterns? Were there any questions where counting forwards would take just as long as counting backwards?

Sample answer: a) to e) counting forwards; g) to k) counting backwards; f ) counting forwards or backwards. Pattern: When the second number is larger than 6, counting forwards is faster. When the second number is smaller than 6, counting backwards is faster. For part f ), counting forwards took just as long as counting backwards.

2. Write a subtraction problem that would take a long time to solve by counting forwards or by counting backwards. Hint: Use 2-digit numbers for both numbers.

Sample answer: 53 − 22

3. Use counting forwards or counting backwards to find the missing number.

a) 9 − = 2 b) 9 − = 6

c) 15 − = 10 d) 15 − = 4

e) 38 − = 6 f ) 38 − = 34

Answers: a) 7, b) 3, c) 5, d) 11, e) 32, f ) 4


unit 7 number sense: addition and subtraction with … for new... · · 2017-06-15copyright ©...

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