Teacher’s Guide for Workbook 2.1
In this unit, students will explore numbers up to 100: they will count (match numbers with their corresponding quantities and numerals), order numbers using different materials (hundreds charts, number lines, place value), represent numbers in different ways (pictures, numerals, tens and ones blocks, number words, and lengths) and compare quantities (more, less, fewer, as many as). They will also learn to add and subtract using different strategies (pictures, number lines, hundreds charts, counting on, counting back, using addition to subtract, and using 10). Students will also begin solving and creating word problems.
Materials Number Cards (0–20) and Number Word Cards (zero–twenty). Write each numeral from 0 through 20 and each number word from zero through twenty on an index card or piece of construction paper. Each student will also need a set of these cards, and you can use BLM Numbers Template (p G-1) to make them. You will use these cards throughout the unit for demonstrations; students will use them as manipulatives (e.g., for sorting and ordering activities, to play Memory). The same numbers, in both forms, should be posted or displayed in the classroom for student reference.
Hundreds Charts and Base Ten Materials. Make a copy of BLM Hundreds Chart (p G-2) 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 G-3) with paper ones and tens blocks from BLM Base Ten Materials (p G-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 G-5).
A Hundreds Chart for Whole-Class Teaching. For whole-class discussions and demonstrations, use a pocket hundreds chart, a hundreds chart poster, or an overhead projector. You could also create a large hundreds chart on the board or on chart paper.
Paper Sticks. Glue 1 cm grid paper (you can use BLM 1-cm Grid Paper (p G-6)) to Bristol board or thin cardboard (e.g., a cereal box). Make paper sticks 1 cm wide of lengths 2 cm, 3 cm,…, 10 cm. As an alternative, if you have Cuisenaire rods, simply add grid markings at each 1 cm mark on one side of the rods. You could do this using a sharp tool, such as scissors. If using Cuisenaire rods, however, be careful not to create false associations between numbers and colours. Students will use these sticks/rods in several lessons, both in Part 1 and Part 2. You will need many copies of these sticks: for some activities, you will need only two of each length per student; for others, you will need six or seven of each length per student (in which case, you might choose to have students work at stations instead).
Dice. Have students make their own “dice.” There are two ways to do this, both of which will be useful in different situations.
Part 1 Number Sense
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1. Use the nets on BLM Cubes (p G-7). You will need to show students how to cut out the flaps correctly if students use glue (if they use tape, this is not as important). Have students write the numbers on the net before folding and making the dice, and ensure that students put the numbers on the outside of the cube.
2. Use plastic or paper egg cartons to mimic rolling dice. Have students bring in 6-pack egg cartons (bring in extras in case students forget). Start collecting the cartons several weeks before you need them in NS2-30. A 12-pack cut in half will also work. Have students write different numbers in each hole in the carton, or write the numbers on paper and tape or glue them to the carton. To mimic rolling two dice, students put two counters into the carton and shake, then open the carton and see which numbers the counters landed on. Make sure that when students shake the carton, they cover up any holes where counters can fall out. Variations: • Write the numbers 4 through 9 instead of 1 through 6 in the carton.
• Use a 12-egg carton to imitate 12-sided dice. • Put three counters in the egg carton to imitate rolling three dice.
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.
Coins or two-colour counters. Two-colour counters will be used repeatedly in this unit. If you don’t have any, play coins (using heads and tails as the two “colours”) can be used instead.
How to make a number line from a hundreds chart. Cut out a hundreds chart (you can use BLM Hundreds Chart or BLM A Larger Hundreds Chart) such that there is 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.
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
B-3Number Sense
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Incorporate Math into Your Daily Classroom Routines You can easily link any of the following activities to the relevant lessons when they are taught.
Line up by number. Have students pick a number card and line up in order according to their numbers. At first, have numbers only go as high as the number of students; later, numbers can go higher than the number of students so that there are gaps in the numbers held by consecutive students.
Refer to students using ordinal numbers. EXAMPLE: Ask the 5th student in line for assistance. Using ordinal numbers throughout the year to call on students will help them learn ordinal numbers (in NS2-19) more easily.
Count back to indicate time remaining. You might count back from 3 to 0 when you need students to quiet down, or count back from 20 to 0 when you want them to line up for lunch. Eventually, end at numbers other than 0. Students who need to finish a task when you say 4 will learn quickly that 5 comes right before 4 when counting back.
If different groups line up at different times (everyone is not getting up together), count back and have one group get ready at 20, another at 15, then 10, and so on to 0. Once students become very familiar with this routine, vary which groups go at a certain number. Later, use different evenly spaced numbers, such as 18, 14, 10, 6, and 2.
Recurring Games The following games and activities recur throughout this unit and others. Rules and materials vary per lesson as students learn more about numbers and counting.
Go to page —. Make sure students can find the page numbers in their JUMP Math workbooks, in the bottom left and right corners. Have students turn to different pages, one at a time, in random order. Always ensure that the entire class has found one page before asking students to turn to another. Have students point to where they see each page number. This helps students grasp the order of numbers, as they learn which way to turn the pages.
Picking pairs. Use, for example, Number Cards and Number Word Cards (see above); the deck that students use will depend on the lesson. Students can play in teams or individually. Place a 3 × 4 array of cards face up on the table. Students take turns picking pairs of matching cards and placing them into a common discard pile. When there are no more pairs in the array, more cards are added to it. The goal is to place all the cards into the discard pile. If students have any non-matching cards left at the end, then some of their cards must have been matched incorrectly.
B-4 Teacher’s Guide for Workbook 2.1
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Memory. Students turn over two cards at a time. If the cards match by number, students set these cards aside; otherwise, they turn them face down again and continue playing. Play this first as a whole class, with volunteers taking turns. Students can then play individually or co-operatively in pairs. In either case, the goal is to finish all the cards. If playing with a partner, Player 1 leads by choosing and turning over a card and Player 2 follows by choosing and turning over another card. After all pairs are found, players switch roles and play again. Players can help each other by asking questions or making suggestions (EXAMPLE: “I think I know where both 3s are; should I turn one of them over?”) but they are not allowed to tell each other where specific cards are. (NOTE: It is a good idea for students to play Picking Pairs—to practise making and recognizing matches—before they play Memory.)
Dominoes. Make paper dominoes with numbers written in different ways (EXAMPLES: random arrangements of dots, base ten blocks, addition or subtraction sentences, numerals). You can use the template on BLM Blank Domino Cards (p G-8). Decide how many different numbers you want the dominoes to have (at least seven for four players), and ensure that every number appears on the same domino with every other number including itself (for four players, there will be at least 28 cards). Explain that the dominoes can be turned around even though any numerals won’t look like numerals any more.
Lay all the dominoes face down and shuffle them. Each player draws a domino in turn. Continue drawing dominoes until all dominoes are taken. The player with either the most dominoes or the highest double (a “double” is a domino with both ends showing the same number) starts the game by laying that domino face up. On a turn, players either (1) play a domino that matches an open end of a domino already in play, or (2) play any domino to start a new train.
At the end of a turn, players may join two existing trains if they wish. (This process can be made more fun by making train sound-effects as the trains are being joined.) The players are a team and must help each other to place their dominoes; all dominoes in each player’s hand are thus placed face up on the table for all to see. The game ends when all dominoes have been played. The goal is for all the dominoes on the table to form a single train. Easier Variation: Play without doubles dominoes (e.g., 5, 5).
Peace. (A co-operative version of the card game War.) Two players sit opposite each other and divide the deck into two equal piles, one on Player 1’s left and one on Player 1’s right. Player 1 begins by turning over the top card of each pile: If the cards are not equal, both cards are placed beside the pile that the greater card came from. If they are equal, they are each placed beside the pile they came from. Player 2 then takes a turn by turning over the top card of each pile. The game ends when all cards have been turned over and played. There will now be two piles on the table. Together, the players must predict, without counting, which pile has more. They count or use one-to-one correspondence to check their prediction. If they are right, they win.
B-5Number Sense
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B-5
Variations: Peace for Less: Place both cards beside the pile that the lesser card came from. Addition Peace: Turn over the top two cards from each pile and compare the sums of each pair.
Difference Peace: Turn over the top two cards from each pile and compare differences instead of sums.
I Have —, Who Has —? Each student needs one card to play (see sample in margin). You can make the cards or have students make them using BLM Game Cards (p G-9). 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 in the margin 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. Early in the unit, when only numbers 1 to 10 are available, students can play in smaller groups. When they have more numbers, students can play in larger groups or even as a whole class. Ready-made cards (on BLMs) are also available for some lessons.
Sample Card I have 3
Who has 000 0 000
Group Dominoes. This is a variation of I Have —, Who Has —? Have one student tape his or her card to the board. The person whose top matches the bottom of the card on the board adds his or her card below it, as when you play dominoes. This variation is particularly useful for students who prefer physical action to verbal answers. You can play with the cards from either Dominoes or I Have —, Who Has —?”
Message booklets. Write one word per page, as shown in the margin. The words should form a sentence, but should be out of order (the first word in the sentence should appear on page 1). Instructions at the top of each page tell students to “Go to page __” to find the next word in a “surprise” message. EXAMPLE: “The pig took a bath in the mud.” would be written over 8 pages. If the words in the sentence appear on pages 1, 5, 3, 6, 2, 4, 7, 8, page 5 would say “Go to page 3” and “pig,” page 3 would say “Go to page 6” and “took,” and so on. As students learn larger numbers, you could make longer books. Variation: Create a 26-page booklet with all the letters of the alphabet in random order but without the “Go to”
B-6 Teacher’s Guide for Workbook 2.1
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instructions. (Make a list of the letters and page numbers for yourself.) Give students oral instructions to create an unlimited number of words and messages. Use messages that appeal to your students’ interests or that relate to class activities. EXAMPLE: “Let’s find out where we’re going on our next field trip. Go to page 5. Now go to page...”
Go to page 5
THE 1
Missing Number Game. Give each student a sheet of paper divided into three equal parts:
Have students write numbers in the first two parts. EXAMPLE:
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:
(sum)
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.
Catch. You will need a small ball or paper object that students can catch in one hand. Throw the ball to a student while saying a number. The student catches the ball with one hand and repeats the number. The student then throws the ball back to you and says whatever “next” number you have asked for (e.g., the next number counting backwards, the next number when skip counting by 5). Ensure that everyone gets a chance to play.
3 3 8
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Meeting Your Curriculum All of the topics covered in this unit are required for students following the WNCP curriculum, either as review or as core curriculum material. The following topics are optional for students in Ontario: creating word problems (NS2-18), solving problems involving missing addends, subtrahends, or minuends (NS2-38 and NS2-39), working with the “not equal” symbol (Workbook page 64).
B-8 Teacher’s Guide for Workbook 2.1
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NS2-1 Counting Page 1
CURRICULUM EXPECTATIONS Ontario: 1m13, 1m20; review, 2m1, 2m5, 2m7 WNCP: 1N1, 1N3; review, [R, CN, V, C]
VOCABULARY the numbers 0 to 10 number how many count
Review saying the numbers from 1 to 10. Teach a counting song, such as “One two, buckle my shoe.”
The concept of how many. Show students sets of four cards from BLM Quantities, of which three illustrate the same quantity, and ask students to identify the card that doesn’t belong. Point to each card, one at a time, and ask students to raise their hands when you point to the card that doesn’t belong. Repeat for each quantity from zero through nine at least once. Discuss what is the same and what is different about all the cards that do belong. Explain that you made the groups based on how many shapes were on each card. It doesn’t matter what the shapes are, how big they are, where they are on the card, or what colour they are.
Tap your desk a few times and ask students to identify the number of taps. Have all students individually hold up the correct number of fingers. Then hold up various number of fingers and have students say the correct number.
Counting in different ways gives the same answer. Arrange nine counters in a row. ASK: Do you think I will get the same answer starting here (at the left) as I get starting over here (at the right)? Count in both directions. ASK: Why did I get the same answer? (same number of counters) Repeat with different numbers of counters. Occasionally make a mistake by counting a counter twice. Wait for students to discover your mistakes. Discuss strategies to ensure that you don’t count objects twice, for example, move objects already counted to a separate pile or cover up each object that has already been counted.
Identifying the numeral with the sound. Draw several capital or lowercase letters and ask students to name them. Explain to students that just as
Goals Students will learn to count and will associate numbers (spoken) with the corresponding quantities and written numerals.
PRIOR KNOWLEDGE REQUIRED
Is able to circle a group of objects Can colour
MATERIALS
BLM Quantities (pp B-104–B-108) 9 counters various old magazines and catalogues (sports, clothing, toys, and so on) packages labelled with numbers BLM 2-cm Grid Paper (p G-10) BLM Game Cards (p G-9) BLM Blank Domino Cards (p G-8)
PROBLEM SOLVING
Number Sense 2-1
we have symbols for the letters in the alphabet (e.g., E is “ee”), we have symbols for numbers. Write some numbers on the board in order, from 0 to 9, and ask students to say the numbers as you point to them. Gradually increase the difficulty by writing more and more numerals that are not in order (4 2 5 3 8 6 1 0 7…). Then write 10 on the board. ASK: Is this a number? (yes) What number is it? (ten)
Identifying the numeral with the quantity. Write the numbers from 0 to 9 across the board, in order, leaving plenty of space between each one. Give each student one of the quantity cards used earlier and ask volunteers to tape their card below the correct number. More than one card will go with the same number. Then write a numeral on the board and have students hold up the corresponding number of fingers.
ACTIVITIES 1–7
1. Five. Give students BLM 2-cm Grid Paper. Ask them to colour any five squares, but only five. Ask one student to count his or her squares, pointing to each square one by one. SAY: I see all of the squares are [describe their arrangement on the page, e.g., in the top corner, in a line]. Did anyone colour five squares in a different way? How is your five different?
2. Posters. Give each student an old magazine or catalogue. Assign each student one number from 2 to 9 and ask students to find and cut out pictures where items are in groups of that many. Students can then form a group with other students who had the same number and pool their cut-outs to make a poster.
3. Numbers on packages. Have students identify the numbers on packages and discuss why numbers are important here. EXAMPLES: puzzle pieces, Lego building blocks, marbles, cookies, pencils, pens, erasers, crayons, chalk, paper, Ziploc bags. Students can also package a product themselves and write how many on the package.
4. I Have —, Who Has —? or Group Dominoes. (See NS Part 1— Introduction) Use numerals on top and dots on the bottom. Alternatively, use different arrangements of dots on the top and bottom. See BLM Game Cards.
5. Dominoes. (See NS Part 1 – Introduction) Use dots on both sides of the dominoes, but arrange the dots differently for the same quantities. See BLM Blank Domino Cards.
6. Finding page numbers. Have students open their JUMP Math workbook to page 1. Then have them turn and point to the following page numbers in order: 2, 5, 3, 7, 10, 6, 9, 8, 6, 1, 4.
7. Message booklet. Make books with 10 pages. Each page has a word or letter and a page number. Give students various messages to find. The same book can be used for several different short messages, as long as the instructions “Go to page…” are given orally.
PROBLEM SOLVING
Real world
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CURRICULUM EXPECTATIONS Ontario: 1m11, 1m13, 1m20; review, 2m3, 2m5 WNCP:1N1, 1N2, 1N3; review, [CN, T]
VOCABULARY the numbers 0 to 10 how many number
Goals Students will practise counting, that is, matching numerals and quantities.
Numbers need to be right side up. Demonstrate that a chair, no matter how you turn it, is still a chair. But letters and numbers are not like chairs; they have to be written “right side up” otherwise they change. Write some lowercase letters, like “j” or “k,” on cards and turn them upside down and sideways to illustrate this. NOTE: Students may identify letters and numbers that don’t change (e.g., 8) or letters that turn into other letters (e.g., “d” becomes “p”) when written on a card and turned upside down. Point out that these are special cases; in general, numbers and letters have only one right side up.
Draw several numbers in two ways, correctly and incorrectly, and have volunteers circle the correct way. Include numbers that are upside down or on their side.
Match by counting. In a two-column chart, draw three different quantities (less than ten) in the first column. Draw the same three quantities, using different items in a different arrangement, in the second column. EXAMPLE: 4 stars, 5 dots, and 1 checkmark in the first column; 1 heart, 4 squares, and 5 triangles in the second column. (Alternatively, use cards from BLM Quantities.) Have volunteers match the items by quantity. Repeat several times, gradually increasing the quantities in each column, up to ten. Then arrange and match quantities by row instead of column. When students can comfortably match quantities, replace the quantities in one column or row with numerals, and have students match numerals to quantities.
ACTIVITY 1
Ask students to walk around the room and look for numbers written the correct way. Have them use the numbers they find to circle numbers written correctly on BLM Circle the Numbers. Some boxes include two correctly written numbers (6 and 9).
PRIOR KNOWLEDGE REQUIRED
Understands the concept of quantity Can join two figures with a line
MATERIALS
BLM Circle the Numbers (p B-109) quantity cards or BLM Quantities (pp B-104–B-108) 2-cm grid paper or BLM 2-cm Grid Paper (p G-10) BLM Game Cards (p G-9) BLM Dominoes (p B-110)
NOTE: Technically, a number is the quantity and the symbol for the number is called the numeral. A digit is any symbol from 0 to 9. A numeral can consist of one digit (e.g., 6, which corresponds to the quantity six) or more than one digit (e.g., 12, which corresponds to the quantity twelve). Students do not need to use the word “numeral” at this stage; they can use “number” to refer to both the quantity and the symbol.
B-11
Number Sense 2-2
Match two quantities to numerals. Ask students to match dominoes with dots to corresponding dominoes with numbers. EXAMPLE:
Encourage students to check both sides of the dominoes they match to verify their answers. Justifying the solution Repeat with other sets of dominoes where each number appears only once. Then begin to include examples where the same number occurs on one side of two different dominoes. Finally, arrange the dominoes in rows instead of columns and then scatter them.
CONNECTION Literature What Comes in 2’s, 3’s, and 4s? by Suzanne Aker One Gray Mouse by Katherine Burton Feast for 10 by Cathryn Falwell One Hungry Monster by Susan Heyboer O’Keefe
Extensions 1. Have students match objects by number. SAY: It might be tricky. Some
groups have the same objects but you have to match by number, not by object. EXAMPLE:
2. Ask students to think of letters that can be turned around to make other letters. Then ask them to think of numbers that can be turned around to make letters. Give students calculators, and have them push different numbers and then turn the calculators around to see what letters they can make. Ask them to try to make a word. Can they make any of these words: hello, goose, giggles, lego, bees? What other words can they make?
5 4
1 3
2 6
PROBLEM SOLVING
BLM Dominoes
EXTRA PRACTICE ACTIVITY 2
Play Picking Pairs and then Memory (See NS Part 1—Introduction) using quantity cards. Start with two of each quantity from one to nine. Arrange the 18 cards in 3 rows of 6. Variation: Use one quantity card and one number card for each quantity.
Draw a group of 9 and a group of 10. Have a partner circle the group of 10.
JOURNAL
ONLINE GUIDE
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CURRICULUM EXPECTATIONS Ontario: 1m11, 1m13, 1m20; review, 2m1 WNCP: 1N5; review, [R, CN]
VOCABULARY more less pair enough as many
Goals Students will identify which of two sets has more by using one-to-one correspondence.
Adding one to both or removing one from both doesn’t change which has more. Take a pile of 3 red counters and a pile of 4 yellow counters. ASK: Are there more red counters or yellow counters? (yellow) Verify by counting. Emphasize that 4 comes after 3, so there are more yellow counters than red counters. Continue adding one to each pile, asking which pile has more, and verifying. Emphasize that adding one to each pile at the same time doesn’t change which one has more.
Matching chairs to people. Sit in your chair and ask students to do the same so that everyone in the classroom is seated. ASK: Are there more people or chairs in this room? How do you know? (If there are extra chairs, then there are more chairs than people.) Draw several combinations of chairs and stick-people on the board (see suggestions below) and ASK: Are there more people or chairs? How do you know? Did you need to count?
• 5 chairs and 7 people; 2 people are standing • 5 chairs and 7 people, but no one is standing—the first two and last
two people are sharing a chair • 9 chairs and 6 people; three chairs are empty
Which group has more? Tell students you want to find out if there are more boys or girls in the class without counting. Ask students to pair up, one boy with one girl. ASK: Are there any boys or girls left without a partner? Are there more boys or girls? How many more?
Find out which pile has more, without counting, by removing one from each pile. SAY: Julie and Teah each have a pile of beads. (Show Julie’s
PRIOR KNOWLEDGE REQUIRED
Understands the concepts of more and less (or fewer) Can count
MATERIALS
lots of objects to count, such as counters and connecting cubes
ACTIVITY
Co-operative musical chairs. Play as you would musical chairs, but no one sits out: Every time a chair is removed, children sit two or more to a chair. Eventually they will all have to fit onto one chair. Play in groups of 7 or 8. Make the connection between having more people than chairs and having to share chairs. VARIATION: Large hula hoops are islands. The water level is rising and islands are disappearing, one by one. People stand inside the hula hoops.
Draw two more hearts than circles. Draw as many pencils as erasers.
JOURNAL
B-13
Number Sense 2-3
pile of 24 yellow counters and Teah’s pile of 26 red counters.) They want to know if they have the same number or not, but counting each pile is too much work. How can they find out without counting? Encourage students to talk over the problem with a partner before sharing ideas with the class. If no one suggests removing one from each pile until only one colour is left, suggest it yourself and then demonstrate. ASK: Which colour is left: red or yellow? (red, so there are more red counters than yellow counters) Who has more counters? (Teah) ASK: If Teah gives a counter to Julie, do you think they will have the same number? Check the prediction. Then let students work in pairs. Give each pair a pile of red and a pile of yellow counters and have them determine if they have more red or yellow counters.
Draw a model for the counters. Draw several squares, some coloured and some uncoloured, scattered on the board. Demonstrate pairing objects by drawing a circle around pairs or by joining pairs with a line. ASK: Are there more coloured or uncoloured squares? How do you know?
Connect one-to-one correspondence with counting. Explain that when you count, you are really pairing up each object with a number. ASK: How many numbers do I say when I count from 1 to 8? (8) Demonstrate by counting 8 cubes. Point out that each cube gets paired up with a number from 1 to 8. Since you know that there are 8 numbers from 1 to 8, there are 8 cubes. Emphasize that it doesn’t matter which cube you pair up with each number, just like it didn’t matter which red counter was paired up with which yellow counter above.
Extension Starsweeper. Before they play this game, students should complete BLM Counting Starred Squares (pp B-111–B-113). Over the course of the BLM, students will learn to identify how many starred squares each square in a grid is touching (see examples in margin).
To make a 4 × 4 or 5 × 5 Starsweeper grid, put at most four stars in the 4 × 4 grid and five stars in the 5 × 5 grid. Then write the number of starred squares each square is touching in that square. You (or your students) can use the templates on BLM Blank Starsweeper Grids (p B-114).
Students cover all the squares on the grid with coins or tokens about the size of a penny. Students remove the coin from any square they think does not have a star in it. If they uncover a square with a 0 in it, they know that all the squares around it are star-free and they can uncover all of those too. When students think there are more starred squares still covered than numbered squares, they stop. Students can check if they’re right by putting the pennies left on the board into two piles: one pile for the pennies that cover a starred square and a second pile for the pennies that cover a numbered square. They win if the first pile has more than the second pile. Students can play individually or co-operatively in pairs by taking turns. Players must decide together when to stop uncovering squares.
Real World
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NS2-4 Counting with a Chart Page 5
CURRICULUM EXPECTATIONS Ontario: 1m11; review, 2m2, 2m5 WNCP: 1N1, 1N3; review, [R, CN]
VOCABULARY number line
Goals Students will use a chart in place of counting orally.
Make a counting strip for each student. Make strips of paper 2-cm wide and 20 cm long divided into ten numbered squares (or photocopy strips from BLM Counting Cubes).
Count using a chart. Give each student up to ten 2 cm connecting cubes (students should have different numbers of cubes). Ask students to count their cubes. Then have them make a chain with the cubes and place it on the chart, so that each cube covers one square and the chain starts on the 1. Students should exchange cubes with different partners and repeat the exercise several times. Then ASK several students: How many cubes did you count? What is the last number covered on the chart? Does anyone notice a pattern? (the last number covered is always the number of cubes in the chain) Then have students repeat the exercise with this pattern in mind. Does the pattern hold? (yes) ASK: What is an easy way to find out how many cubes there are without counting? (look at the last number covered)
The chart does the counting for you. ASK: How is the chart doing the counting for you? (instead of saying “one, two, three,…” when picking up the cubes, just place a cube on 1, another cube on 2, another on 3, and so on) Demonstrate by picking up a cube, saying “one,” and placing it on the 1. Pick up another cube, say “two,” and place it on the 2. Repeat until all the cubes are counted.
The chart makes sure that each cube is counted once. ASK: How does the chart help to make sure that you don’t count any cube twice? (once a cube is placed on the chart, it’s been counted) How does the chart help to make sure you don’t miss any cubes? (if any cubes are left off the chart, they aren’t counted)
PRIOR KNOWLEDGE REQUIRED
Can say the numbers from 0 to 10 and write the corresponding numerals in sequence Can count to 10
MATERIALS
counting strips (details below) or BLM Counting Cubes (p B-115) lots of 2-cm connecting cubes two-colour counters or coins precut square pieces of paper (details below)
1 2 3 4 5 6 7 8 9 10
PROBLEM SOLVING
Making and investigating conjectures
Number Sense 2-4
Demonstrate using the chart incorrectly. Draw the same chart on the board and use square pieces of paper to represent cubes. Place six squares on numbers as shown:
Explain to students that because 8 is the last number covered, you think that you put 8 squares on the chart. ASK: Am I right? (no) Why not? (the squares must cover every number in order; you can’t skip numbers) Then take the squares off and demonstrate counting them incorrectly: 1, 3, 4, 5, 7, 8. SAY: Even when I count them, I still get 8. What did I do wrong now? (you missed two numbers; you didn’t say all the numbers in order) Explain that just as you’re not allowed to miss numbers when counting, you’re not allowed to miss any numbers when using the chart. Repeat with various incorrect placements, always asking students to tell you how this is like missing numbers when counting. EXAMPLE: 2, 3, 4, 5.
Writing numbers. From this lesson forward, students need to be comfortable writing the numerals from 0 to 9.
Extension On BLM Counting Dots (p B-116), students can count the corners (marked by dots) of various shapes.
1 2 3 4 5 6 7 8 9 10
PROBLEM SOLVING
Connecting
ACTIVITY
Give each student 10 two-colour counters or coins. Have students toss the counters/coins and then use a sequence of numbers to count how many turned up red and how many turned up yellow (or heads and tails). Students could place the red counters (or heads) above the row and the yellow counters (or tails) below the row.
1 2 3 4 5 6 7 8 9 10
Students can practise writing numbers with BLMs Ants, License Plates, Roman Numbers, and Writing Numbers.
ONLINE GUIDE
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NS2-5 More, Fewer, and Less Page 6
CURRICULUM EXPECTATIONS Ontario: 1m11, 1m20; review, 2m1, 2m7 WNCP: 1N5, 1N8; review, [R, C]
VOCABULARY right left more most less least fewer fewest order
Goals Students will understand that the number that means more (less) is said later (earlier) when counting and written to the right (left) when the numbers are written in order.
The concept of more. Ask students to try to explain what “more” means without using the word. Then explain that “more” in math means a larger number. Write “more” on the board. Show lots of pennies in one hand and two or three in the other. ASK: Which hand has more pennies? Then draw lots of little circles on the right side of the board and two big circles on the left. ASK: Are there more circles here or there? Explain that the circles are bigger on one side, but there are more of them on the other side.
The number you say last means more. Show two piles of blocks, one with 8 and one with 9. ASK: Which pile has more blocks? How can we find out for sure? Then count the pile with 8 blocks. Choose a student who said that the pile with 9 blocks has more. SAY: You said that the other pile has more. Do you think I will get to eight when I count the second pile? Emphasize that you should get to eight before finishing the second pile because it has “more.” Then count together and stop at eight. ASK: Was [student’s name] right? Explain that because you were not finished counting the other pile when you said eight, that pile has more.
Show 5 red counters and 7 yellow counters. Count the pile of seven and then check to see if you say seven when you count the other pile. ASK: Are there more red counters or yellow counters? (yellow) How do you know? (when counting the red pile, you didn’t say the number that you got when you finished counting the yellow pile)
It’s easier to count two piles together. SAY: It’s so much work to count each pile separately; let’s try to count two piles at the same time. Show a pile of 6 red cubes and 8 yellow cubes. Taking one of each colour at a time, count up to six; hold up 1 red cube and 1 yellow cube with each number. Explain that you have to stop because you have run out of red cubes. Since there are extra yellow cubes, you know there are “more” yellow than red cubes. Write on the board: red 6. Finish by counting the two extra yellow cubes. Emphasize that you can start at 7 because you already counted 6.
PRIOR KNOWLEDGE REQUIRED
Is able to say the numbers from zero to ten in sequence Can match, and translate between, numbers spoken orally and numerals
MATERIALS
blocks, counters, cubes or other objects to count BLM Who Is Winning? (p B-117)
B-17
Number Sense 2-5
Then write on the board: yellow 8. Give students red and yellow cubes to count in this way. Repeat by having students trade handfuls of cubes with each other.
When numbers are written in order, the number on the right means more. Write the numbers in order on the board. ASK: Are the numbers written in the same order as you say them when counting out loud? (yes) How could you use this order to say if a number is more or less than another number? (the one on the right, or further along in the list, means more, just as the number you say last when counting out loud means more)
Which number means more? Write two numbers on the board. Have students show the larger of the two numbers by holding up the correct number of fingers. Have an ordered list of numbers displayed for reference. Eventually challenge students to indicate which is more without referring to an ordered list.
Which number means the most? Explain that “most” means more than all the others. Write “most” on the board. Write three numbers on the board and have students choose the number that means the most. Start with examples where the numbers are already in order (EXAMPLE: 3, 6, 7), and then give examples where the numbers are not in order (EXAMPLE: 7, 4, 1). Students might at first find it helpful to refer to a list of the ordered numbers. They can circle all three numbers that they are asked to consider on the list and then choose the one furthest right as the most.
Introduce “fewer” and “less” as the opposite of “more.” Have two piles of counters: 5 red and 3 yellow. Tell students there are more red counters than yellow counters; that means there are fewer yellow counters than red counters. Explain that fewer is used for amounts that you can count and less is used for amounts that you cannot count. Show or draw two students with different amounts of cake: One has 2 small pieces, the other has 1 large piece bigger than both small pieces put together. ASK: Who has more pieces? Fewer pieces? More cake? Less cake? Write “fewer” and “less” on the board, spaced apart, and ask students to point to the correct word to finish various sentences (or make cards for the students to hold up). EXAMPLE: I have more coins, so you have coins. (fewer) Repeat with: carrots (fewer), juice (less), pie (less), pizza (less), pieces of pizza (fewer).
Repeat this lesson with “fewer/less.” Go back to “The number you say last means more,” and guide students to decide which pile has fewer by first asking which pile has more. Introduce “least” and “fewest” as the opposite of “most.” Explain that least means less than all the others and fewest means fewer than all the others. Repeat the last exercise above (Which number means the most?) with “fewest” instead of “most”.
PROBLEM SOLVING
BLM Who Is Winning?
Doing a simpler problem first
Bonus Give students 4 blue, 8 red, and 7 yellow cubes and ask them to count all three piles by saying the counting sequence only one time.
Extension—Introduce the more than (>) and less than (<) symbols using BLM Mr. Hungry.
ONLINE GUIDE
ACTIVITY
Play Peace and Peace for Less. (See NS2 Part 1—Introduction) Use only the red cards from A to 10 and count A as 1.
Bonus 7, 6, 3, 9; 4, 6, 2, 3, 7, 1
B-18 Teacher’s Guide for Workbook 2.1
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NS2-6 How Many More? Pages 7-9
CURRICULUM EXPECTATIONS Ontario: 1m11, 1m14; review, 2m3, 2m5, 2m6, 2m7 WNCP: 1N5; review, [R, CN, C, V]
VOCABULARY extra pair up how many more
Goals Students will determine how many more by pairing objects up and counting the extras.
Count the extras to find how many more. Give students two-coloured counters to toss. ASK: Did more counters land with the yellow face up or the red face up? Have students pair up their counters to see which colour has extras. ASK: How many extras are there? Have several volunteers present their answers, showing their pairings. Repeat several times.
Find out how many more by lining up objects above and below a sequence of numbers. Draw the numbers 1 to 10 on the board, then line up eight squares above the numbers and six triangles below the numbers in one-to-one correspondence:
1 2 3 4 5 6 7 8 9 10
Remind students how to pair objects, one square to one triangle. ASK: Are there more squares or triangles? (squares) SAY: If there is more of one shape, I’m going to call the additional number of shapes “extra.” Write the word “extra” on the board. Draw a circle around each extra square and the number below it:
1 2 3 4 5 6 7 8 9 10
ASK: How many extra squares are there? (two) Write the following sentence on the board and ask a volunteer to fill in the blank: There are more than . Repeat with similar pictures.
Counting the extra numbers you say. Write 1 2 3 4 5 and have a volunteer continue writing the numbers until 8. ASK: How many extra numbers did you write? (3) How many more is 8 than 5? (3) Tell students that they can keep track of how many extra numbers there are by counting on their fingers. Tell students you are going to count to 8, but only raise a finger when you say an extra number after 5. Remind students that you want to know how many more 8 is than 5. Count from 1 to 5 with your fist closed,
PRIOR KNOWLEDGE REQUIRED
Understands one-to-one correspondence Understands the concepts of more and less (fewer) Can count
MATERIALS
BLM Counting On (p B-118) BLM How Many Fruits? (p B-119)
Making a model.
Number Sense 2-6
then raise your thumb and say “6,” raise your index finger and say “7,” raise your middle finger and say “8.” SAY: Because I raised 3 fingers when counting to 8 after I counted 5, I can see that 8 is 3 more than 5.
As a class, use this method to find how many more 9 is than 7. Start counting at 1; students only raise fingers when they get to the extra numbers. Repeat with 8 and 4; 10 and 5; 9 and 6; 10 and 7.
NOTE: Make sure students tuck their thumbs under their other fingers when they make a fist. If the thumb is not tucked under and sticks out, students may start counting the extras with their other fingers but include the thumb when they total the extras. To ensure that students keep their fists closed while saying the first number, you can pretend to throw them the first number which they have to pretend to catch.
Counting on. Show students an easier way to find how many more 10 is than 7. Instead of saying 1, 2, 3, 4, 5, 6, 7, all with their fist closed, they can just say 7 with their fist closed, and count the extra numbers 8, 9, and 10. Discuss why this works. SAY: You are going to get to 7 anyway, by saying all the numbers from 1 to 7, so you might as well save time by starting at 7. Give students lots of practice with this type of question. Eventually include questions where students need to count the extra numbers on both hands, but use only one-digit numbers. EXAMPLE: 9 is how many more than 3?
Counting on with pencil and paper. Tell students that you want to know what number is 4 more than 5. Instead of saying the next four numbers, you can write them. Write on the board: 5 (as on Workbook page 8). Have a volunteer fill in the blanks. ASK: What number is 4 more than 5? Repeat with other numbers, always ending with at most 10. Then write the numbers from 1 to 20 in order on the board, and include problems that require counting to 20. Leave this number sequence visible while students complete Workbook page 8.
Extensions 1. BLM More Than (p B-120). Students discover patterns by changing the
order of numbers: 7 is 4 more than 3 but 7 is also 3 more than 4.
2. BLM Keeping Score (p B-121) shows various scores for Red against Blue and asks who’s winning and by how many points.
Reflecting on other ways to solve a problem
PROBLEM SOLVING
ACTIVITY
Bring in a bed sheet and set up a hiding area at the front of the room. Ask 4 volunteers to hide behind the sheet and ask for 3 more volunteers to stand at the front. ASK: How many children are at the front of the room? SAY: I know there are 4 children hiding even though we can’t see them, so we will count the others starting at 5. Demonstrate doing this and then remove the sheet and count all the students, starting at 1. Repeat with various numbers of volunteers. VARIATION: Hide a known number of counters in a container.
Literature—More, Fewer, Less by Tana Hoban
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CURRICULUM EXPECTATIONS Ontario: 1m12; review, 2m3 WNCP: 1N4; review, [R]
VOCABULARY one, two, …, ten
Goals Students will read the number words from zero to ten.
Sound out number words to read. On the board, write:
two four zero three five one
SAY: These are the number words for 0, 1, 2, 3, 4, and 5, but they are out of order. Write the numbers on the board. Have students say each number out loud. Use sound to match the numerals to the number words in this sequence:
• 4 What sound does it start with? What other words start with the same sound? What letter makes that sound? What sound does the word “four” end with? What letter do you think it ends with? Can you choose the correct word? (When chosen, circle the word “four.”)
• 0 Repeat the questions above. Circle “zero.” Show students how to check their choice using information given. ASK: The word that you circled has an “r” in it—does this make sense?
• 5 There are two ways to see that “five” is 5: first, it’s the only word left that begins with the “f” sound; second, look at all the words in the list and see that “five” is the only one that has a “v” sound as well as an “f” sound.
• 3 Remind students that sometimes two letters make one sound. Ask them which two letters are making one sound in words like throw, thanks, and think. Encourage students to search for the words in a book, point to words on the word wall, or write some of them on the board. Underline the “th.” ASK: Which number word starts with “th”?
• 2 It starts with “t” but not “th.”
• 1 It has an “n” sound; also, it’s the only word left!
Repeat with the words “six” through “ten.” Use the “t” sound at the end of “eight” to help students match it to 8.
PRIOR KNOWLEDGE REQUIRED
Can write the alphabet Knows the sounds associated with each letter of the alphabet
MATERIALS
BLM Match Pictures to Number Words (p B-122) number word cards for zero to ten (one per student) number cards for 0 to 10 (one per student) BLM Reading Numbers (p B-123)
Reflecting on the reasonableness of an answer
PROBLEM SOLVING
PROBLEM SOLVING
EXTRA PRACTICE
Number Sense 2-7
Find the number word in a sentence. Write the number words from “zero” to “five” on the board and then the sentence, “Four friends played together.” ASK: Can you find the number word in that sentence and say it? Ask a volunteer to write the number above the number word:
4 Four friends played together.
Repeat with several more sentences, using “zero” to “five.” Then erase the number words on the board and have students find the number word without the list to refer to. Continue with sentences using number words “six” to “ten,” again starting with a list on the board and then erasing it. Finally, give students sentences using any number from “zero” to “ten.” Start with simple sentences, such as “There are nine monkeys,” and move on to more complex sentences, such as “Rita bought two tennis rackets and three tennis balls.” EXAMPLES:
Four children played hockey. Recess lasts ten minutes. Rita bought three tennis balls. Mary has five erasers. John is seven years old. Karen is five years old. Calli is three years old and Lina is five years old. John has eight fingers and two thumbs. Lucas is two years younger than Sarah.
Ask students to make up their own sentences and have a partner write the number(s) above the number word(s).
A tip for struggling students. When you ask students to write the numbers above the number words, here and on Workbook page 11, some students may find it helpful if you underline the number word first. Once students are able to find and write the number this way, try more sentences without underlining the number word, or photocopy Workbook page 11 and have students redo it.
Extensions 1. BLM How Many More Than (p B-124). Students write how many more
one number is than another.
Bonus Add the number word above the numeral in the blank.
2. BLM Stars (p B-125). Students join the dots in order, according to the number words.
3. Give each student number word cards for “zero” to “ten.” Students shuffle the cards and order them. Students can then re-shuffle the cards and exchange with a partner.
Representing
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NS2-8 Addition Pages 12-13
CURRICULUM EXPECTATIONS Ontario: 1m25; review, 2m1, 2m2, 2m5, 2m6, 2m7 WNCP: 1N9; review, [R, V, CN, C]
VOCABULARY add plus (+) in total altogether equal (=) addition sentence
Goals Students will solve simple addition problems.
Starting with 2 and adding 3 more always gives 5 in total. Draw two circles in a row on the board. ASK: How many circles did I draw? Then ask your students to watch carefully. Draw three more circles. ASK: How many more did I draw? SAY: I started with two circles. I drew three more. How many do I have in total? Repeat with squares in a row and then triangles arranged not in a row, again starting with two and adding three more.
The plus (+) and equal (=) signs. ASK: If you had two apples and someone gave you three more apples, how many would you have in total? Tell students that mathematicians have a way to say that if you have two of something and you add three more, you always have five in total. Ask if anyone knows the way mathematicians write this. If no one does, write 2 + 3 = 5. Ask if students know the way mathematicians say this. Tell them that we say “2 plus 3 equals 5” but what we really mean is “starting with 2 things and adding 3 more is the same number as having 5 things”; point to the corresponding symbol as you say each part. Emphasize that the plus sign (+) means “adding” and the equal sign (=) means “is the same number as.”
Read addition sentences two ways. Write 3 + 4 = 7 on the board. ASK: How could I read this? (“3 plus 4 equals 7” or “starting with 3 things and adding 4 things is the same number as having 7 things”) Say it both ways after volunteers respond. Repeat with more sentences, but don’t include zero yet (students will add and subtract zero in NS2-10). EXAMPLES: 2 + 1 = 3, 2 + 4 = 6, 1 + 5 = 6, 3 + 3 = 6, 4 + 5 = 9, 3 + 5 = 8.
Check with counters that addition sentences are right. Give students two-colour counters or two colours of blocks. Have students make, for example, a pile of 2 yellow counters and another pile of 4 red counters and then see how many they have altogether. Emphasize that starting with 2 counters and then adding 4 more counters is the same number as having 6 counters (i.e., starting with both piles put together). SAY: Notice that we are adding counters, not colours; colour doesn’t matter. Write on the board:
PRIOR KNOWLEDGE REQUIRED
Uses one-to-one correspondence when counting Can count to 10 Know the plus (+) and equal (=) signs Understands the concept of addition
MATERIALS
two-colour counters or two colours of blocks dice BLM Game Cards (p G-9) BLM Blank Domino Cards (p G-8) BLM Add the Dots (p B-126)
Looking for a pattern
ONLINE GUIDE
Number Sense 2-8
2 + 3 = 5, 3 + 5 = 7, 5 + 4 = 9. Challenge students to find the incorrect sentence and prove that it is incorrect using their counters (when the piles of 3 and 5 counters are put together they do not total 7).
Write the total on the left. Tell students that when you say two things are the same, it doesn’t matter which you say first. For example, “My shirt is the same colour as your crayon” and “Your crayon is the same colour as my shirt” mean the same thing. We can do that with numbers too. Saying 5 + 1 is the same number as 6 (write 5 + 1 = 6 on the board as you say this) means the same thing as saying 6 is the same number as 5 + 1 (write 6 = 5 + 1 on the board). Have students write these addition sentences with the total on the left: 3 + 4 = 7, 2 + 6 = 8, 1 + 4 = 5.
Write on the board: 6 = 3 + 2, 7 = 2 + 5, 8 = 7 +1. Again have students find the incorrect sentence and prove their choice using counters.
Add 3 things together. Tell your students that 3 girls, 2 boys, and 2 adults went on a picnic. Write on the board: 3 girls + 2 boys + 2 adults = people. ASK: How many people went on the picnic? Have one volunteer draw the 3 girls, another volunteer draw the 2 boys, and another draw the 2 adults. ASK: How many people are drawn altogether? Have students find the totals in more such problems by drawing their own pictures or by using counters. EXAMPLES:
3 basketballs + 2 volleyballs + 2 soccer balls = balls vehicles = 3 buses + 2 fire trucks + 3 police cars
Bonus animals = 2 lions + 1 bear + 3 cats + 2 dogs + 1 hamster
Write addition sentences another way. Explain that addition sentences can be written up and down too (see margin). Have students practise adding vertically with more problems like those above.
BLM Add Roman Numbers shows playing cards that use Roman numbers. Students use the cards to write and add roman numbers.
BLM I Have — Who Has — Addition Cards has ready- made cards for numbers up to 5. The BLM has 12 cards: the first 6 go together and the next 6 go together. Play in groups of six.
ONLINE GUIDE
ONLINE GUIDE
ACTIVITIES 1–3
1. If students can count to 12, have pairs of students roll two dice—one each—and add the numbers they roll. Students should add the numbers independently and compare their answers. If students’ answers do not agree, they should add again or count the dots until they do. If students can count to 18, have them work in groups of three.
2. I Have —, Who Has —? (See NS Part 1—Introduction) Use a number on top and an addition question with a picture on the bottom. Use BLM Game Cards to make cards for numbers up to 10.
3. Dominoes or Group Dominoes. (See NS Part 1—Introduction) Use BLM Blank Domino Cards to make dominoes with a number on top and an addition problem with a picture on the bottom.
Drawing a picture
BLM Add the Dots
EXTRA PRACTICE
Literature—Animals on Board by Stuart J. Murphy. Two trucks of each kind of animal pass by the character’s truck and he adds the numbers together to find the total.
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NS2-9 Subtraction Pages 14-15
CURRICULUM EXPECTATIONS Ontario: 1m25; review, 2m2, 2m6 WNCP: 1N9; review, [R, V]
VOCABULARY minus (-) take away subtract subtraction sentence
Goals Students will understand subtraction as “taking away” and will draw and use pictures to solve subtraction problems.
Taking away 3 objects from 5 always leaves 2 objects. Draw five circles in a row on the board. SAY: I want to remove three circles, but instead of erasing them, I am going to cross them out; please watch carefully and tell me to stop when you think I’ve crossed out enough circles. Cross out the first three circles. If students don’t tell you to stop, ASK: How many have I crossed out? Have I crossed out enough? How many are left? Repeat with five squares in a row, but this time take away the last three squares. Then repeat with five triangles scattered randomly and take away any three. ASK: If you had five apples and someone took three of them away, how many would be left?
The minus sign (−). Explain that if you have five of something and you take away three of them, you always have two left. ASK: Does anyone know how mathematicians write this fact using numbers and signs? Encourage students to come to the board to show you if they want to. If no one volunteers, write 5 - 3 = 2. Ask if students know the way mathematicians say this. Tell them that we say “5 minus 3 equals 2,” or “5 take away 3 equals 2,” or “subtract 3 from 5 to get 2.” Point to the corresponding sign as you say each part. SAY: What we really mean is that when we start with five things, and we take away three of them, we get the same number as if we’d started with only two things.
Write subtraction sentences from a picture. Draw seven circles and tell students you want to remove four. SAY: Tell me when to stop. (Cross out four circles.) Ask a volunteer to write a “take away” sentence on the board for your drawing. (7 - 4 = 3) Write “take away,” “subtract,” and “minus” on the board. Ask another volunteer to read the sentence in two different ways, one using “take away” and another using a different word that means the same thing. Repeat this several times with different numbers, asking students to write the sentence and then read it using “subtract” or “minus.” Do not include examples with 0 yet (students will subtract with 0 in the next lesson).
PRIOR KNOWLEDGE REQUIRED
Uses one-to-one correspondence when counting Can count to 10 Knows the plus (+) and equal (=) signs Understands the concept of addition
MATERIALS
PROBLEM SOLVING
Number Sense 2-9
Colouring to subtract. Tell students that instead of crossing out circles, you will colour the circles you want to take away and then ask how many are not coloured. Draw on the board the picture shown in the margin. ASK: How many circles did I draw? How many did I colour? How many are not coloured? (Write 5 - 3 = 2.) Repeat for various examples.
Draw a picture to solve the subtraction sentence. Write a subtraction sentence on the board, such as 5 - 2. SAY: Please draw shapes, as I’ve been doing, to show 5 - 2. You might draw circles, squares, triangles, or hearts. Your shapes should be big enough so that the whole class can see them when you hold them up. Have volunteers show their work to the class; emphasize how all the drawings are different and how they are the same. Differences may include shapes drawn, size of shapes, where they are on the page, and colour.
Check with counters that subtraction sentences are right. Give students counters. Have students count out 7 counters, and ask them to take away 3, see how many they have left, and write the subtraction sentence. Repeat for other examples. Then write these subtraction sentences on the board: 8 - 2 = 6, 8 - 3 = 4, 7 - 5 = 2. Challenge students to find the one that’s wrong and to prove it wrong using their counters.
Write the difference on the left. Emphasize that 7 - 3 = 4 (7 take away 3 is the same number as 4) means the same things as 4 = 7 - 3 (4 is the same number as 7 take away 3). Then have students again use their counters to find the incorrect subtraction sentence among these choices: 5 = 9 - 4, 7 = 9 - 2, 3 = 8 - 4.
Another way to write subtraction sentences. Explain that, like addition sentences, subtraction sentences can be written up and down instead of side to side (see margin). Write several subtraction sentences on the board for students to solve using a picture or counters.
Extension
PROBLEM SOLVING
Drawing picture
EXTRA PRACTICE
BLM Subtract Using Dominoes—students write subtraction sentences for pictures of dominoes.
ONLINE GUIDE
1. Play Difference Peace. (See NS Part 1—Introduction)
2. a) Give students dominoes. Have pairs play as follows: Player 1 picks a domino, counts the total dots, tells Player 2 how many dots are on the domino, and hides one half. Player 2 guesses how many dots are on the hidden half. Player 1 then reveals the hidden half. Players switch roles. b) Play Missing Number Game (see NS 2 Part 1—Introduction) but have students draw dots on the first two flaps and write the total number of dots on the third flap. Students might find it helpful to think of the first two flaps as a domino, with the total number dots given on the third flap.
10 cats - 3 cats = 7 cats
B-26 Teacher’s Guide for Workbook 2.1
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NS2-10 Adding and Subtracting 0 Page 16
CURRICULUM EXPECTATIONS Ontario: 2m2, 2m6, 2m7, 2m72 WNCP: 2N8, [R, V, C]
VOCABULARY add plus minus take away subtract addition sentence subtraction sentence
Goals Students will solve simple addition and subtraction problems involving zero.
Write addition sentences with 0 using dominoes. Tape a large paper domino on the board with a 5 on one side and blank (0) on the other side. Ask students to count the number of dots on each side. SAY: I would like to write an addition sentence for the total number of dots on this domino. Remind students that there is a number that means none (0). Invite answers. Write 5 + 0 = 5 under the domino. Tape a second domino on the board or add dots to the first to show 7 on one side and 0 on the other side. Have a volunteer write the corresponding addition sentence.
Give each student several dominoes, real or paper, that are blank on one side. (You can use BLM Blank Domino Cards to make them.) Have students record the number sentences for their dominoes. ASK: What do you notice? Explain that when students add 0 objects, they don’t add anything, so the result is the same as the number they started with.
Practise adding 0 without dominoes. ASK: If I start with 3 things and add 0 things, how many do I have in total? (3) Write the corresponding addition sentence on the board: 3 + 0 = 3. Repeat with more addition statements. EXAMPLES: start with 5 things and add 0 things; start with 2 things and add 0 things. Invite volunteers to write the corresponding addition sentences on the board: 5 + 0 = 5, 2 + 0 = 2. ASK: What if I start with 0 things and then add 3 things? Now how many do I have? (3) Have a volunteer write the addition sentence on the board: 0 + 3 = 3. Continue with more such questions. Then mix questions with 0 as the first number or the second number. ASK: What do you think 0 + 15 will be? Repeat with 12 + 0, 0 + 18, 10 + 0.
Bonus Use increasingly larger numbers: 20 + 0, 0 + 55, 100 + 0.
Subtract 0 using pictures. Draw 3 circles on the board. ASK: How many circles do I have? (3) Write 3 underneath the circles. ASK: If I want to take away no circles or 0 circles, how many circles would I have left? (3) Count how many are left when no circles are taken away and write the subtraction
PRIOR KNOWLEDGE REQUIRED
Uses one-to-one correspondence when counting Can count from 0 to 10 Knows the plus (+), minus (-), and equal (=) signs Understands the concepts of addition and subtraction
MATERIALS
pre-made paper dominoes (see below) BLM Blank Domino Cards (p G-8) BLM Game Cards (p G-9)
PROBLEM SOLVING
Number Sense 2-10
sentence under the circles, starting with the 3 already written on the board (3 - 0 = 3). Repeat with 5 circles and 1 circle, taking away 0 circles each time. Have volunteers write the subtraction sentences underneath the drawings: 5 - 0 = 5, 1 - 0 = 1. Explain that when you take zero things away, you are left with the number you started with.
Practise subtracting 0 without using pictures. ASK: If I start with 8 things and 0 things are taken away, how many things are left? (8) Have a volunteer write the subtraction sentence on the board: 8 - 0 = 8. Repeat with more subtraction statements. EXAMPLES: start with 7 things and take away 0 things; start with 2 things and take away 0 things. Have volunteers write the subtraction sentences on the board.
Write subtraction sentences that equal 0 using pictures. Draw 7 circles on the board. ASK: How many circles do I have? (7) Write 7 underneath. SAY: I want to take away 7 circles. Then draw an X through all 7 circles. ASK: How many circles do I have left? (0) Write the subtraction sentence under the circles, beginning with the 7 already written: 7 - 7 = 0. Draw 2 circles on the board and cross out 2 circles. Have a volunteer write the subtraction sentence underneath: 2 - 2 = 0.
Practise subtracting without drawing circles. ASK: If we start with 4 things and take away 4 things, how many things are left? (0) Write the subtraction sentence on the board: 4 - 4 = 0. Repeat with 6 things take away 6 things, then 9 things take away 9 things.
Bonus 47 - 47; 312 - 312.
Subtracting with 0. Have students write the subtraction sentences for pictures in which either all the objects are crossed out (0 is the difference) or none are crossed out (0 is the subtrahand, the number being subtracted).
Extension Another model for subtracting. Draw the model shown in the margin on the board, with the corresponding subtraction sentence. ASK: What number is being subtracted, the shaded part or the white part? (shaded) How would you show 6 - 5 = 1 using this model? Have volunteers draw models for 5 - 2 and 4 - 1. Then have students draw models for 6 - 6 = 0, 4 - 0 = 4, 10 - 10 = 0, 10 - 0 = 10. Instead of crossing out (as with the circles), they should shade the number being subtracted.
BLM I Have — Who Has — Subtraction Cards has ready-made cards for numbers up to 5. The BLM has 12 cards: the first 6 go together and the next 6 go together. Play in groups of six.
ONLINE GUIDE
ACTIVITIES 1–2
1. I Have —, Who Has —? (See NS Part 1—Introduction) Use a number on top and a subtraction question with a picture on the bottom. Use BLM Game Cards to make cards for numbers up to 10.
2. Dominoes or Group Dominoes. (See NS Part 1—Introduction) Use cards with a number on top and a subtraction problem with a picture on the bottom. Include 0 in the problems. See BLM Blank Domino Cards.
7 - 3 = 4
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NS2-11 Counting to 20 Pages 17-18
CURRICULUM EXPECTATIONS Ontario: 1m11, 1m20; review, 2m2, 2m3, 2m5, 2m7 WNCP: 1N1, 1N3; REVIEW, [R, C, CN, V]
VOCABULARY numbers to 20
Goals Students will count to 20. Students will learn to keep track of their counting so that they can locate any mistakes and verify their answers with others.
The numbers 14, 16, 17, 18, and 19. Write or tape the numbers from 1 to 20 all in a row and demonstrate counting to 20, pointing to each number as you say it. Then circle the numbers 14, 16, 17, 18, and 19 and ask students to listen carefully as you say them. Then underline the ones digits of those numbers and tell students to listen for those number words as you count again, pointing to each number as you say it. ASK: Do you notice a pattern in how I say those numbers? (the second digit is said first, then the word “teen”) Point to these five numbers in random order and have students say the numbers as you point to them. Then include the numbers from 1 to 10, in random order.
13 and 15. Write these numbers on the board. ASK: Do you know how to say these numbers? Explain that “13” is not “three-teen” but is something close—“thir-teen.” Also, 15 is not “five-teen,” but “fif-teen.” Repeat the exercise above with these numbers included.
11, 12, and 20. Write these numbers on the board. ASK: Do you know how to say these numbers? Explain that these numbers are the hardest to remember because they don’t sound like any number students already know. Teach students how to say these numbers, and repeat the exercise above. Start by focusing only on these numbers, then include all numbers from 11 to 20, and then all numbers from 1 to 20. End by saying the numbers in order from 1 to 20 together as a class.
Count concrete objects. Give each student 4 or 5 cubes to count. Pair up students and ask them to count how many cubes they have together. Then pair up the pairs and ask them to count how many cubes their group of 4 has altogether.
Count objects on paper. Hand out cards with different numbers of objects on them. Ask students to write the number on each object as they count.
PROBLEM SOLVING
MATERIALS
4 or 5 cubes for each studentnumber cards for 1 through 20 cards with different numbers of objects on them BLM Count the Letters (p B-130) BLM Numbers Template (p G-1)
B-29
Number Sense 2-11
Explain that this helps keep track of objects already counted and objects that still need to be counted.
Counting on from 10. Draw a basket with 10 apples in it and then draw another apple outside the basket. Count the apples in the basket as a class, then SAY: There are 10 apples in the basket and 1 apple outside the basket. How many apples are there altogether? Emphasize that there is 1 more than 10 apples, so the number of apples is the number that comes right after 10. ASK: What number comes right after 10? (11) Count all the apples together to verify that there are 11. Repeat with 12 apples and 13 apples. Then SAY: 11 is 1 more than 10, 12 is 2 more than 10, and 13 is 3 more than 10. Write 11, 12, and 13 on the board and point to the ones digit as you say how many more than 10. What number do you think is 4 more than 10? (14) Have a volunteer write the number on the board. Draw a basket with 10 apples and 4 more apples outside the basket and count the apples to verify that there are 14 apples altogether. Repeat with 15, 16, and so on up to 20. Then draw pictures with varying numbers of apples outside the basket and ask students to count the apples outside the basket and then determine how many in total without counting. EXAMPLE: “There are 10 apples in the basket and 7 apples outside, so there are 17 apples altogether.” Write the corresponding addition sentence on the board vertically:
Then have students write the answers to more such addition sentences.
Draw pictures like those on Workbook page 18, with one group of 10 and several other objects, and have students count the objects not part of the 10 to say how many there are in total.
Finally, just write vertical addition sentences and have students find the answer. EXAMPLES:
Literature—So Many Cats by Beatrice Shenk de Reigners. Students can count the cats in each part of the story.
CONNECTION
(Instructions for all Activities are in NS Part 1—Introduction.)
1. Play Picking Pairs and then Memory. Use cards numbered 11 through 20 (see BLM Numbers Template) and cards with 11 through 20 pictures or stickers on them. Make 4 rows of 5.
2. Finding page numbers. Have students open their JUMP Math workbook to page 1. Then have them turn and point to the following page numbers: 7, 13, 10, 16, 19, 8, 6, 14, 15, 17, 2, 5, 3, 9, 4, 6, 18, 12.
3. Message booklet. Make books with 20 pages. Each page has a word or letter and a page number. Give students various messages to find. The same book can be used for several different short messages, as long as the instructions “Go to page…” are given orally.
PROBLEM SOLVING
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Count letters. Write “the” on the board. Have a volunteer count the number of letters in this word. Then write “the mouse” and demonstrate counting the letters, starting at 1. ASK: Is there an easier way to count all the letters? Is there a way to take advantage of the fact that someone already counted the letters in the word “the”? Challenge the students, working in pairs, to think of a solution. (Since “the” has 3 letters, start at 4 when counting “mouse.”)
Now write on the board: T h e m o u s e a t e t h e a p p l e s.
SAY: I’d like to know how many letters are in the whole sentence. Start by counting the letters in “The” and write 3 just above the end of the word. Continue by counting the letters in “mouse” and write the total 8 on top. Remind students that you are counting each letter from the beginning of the sentence. Ask a volunteer to continue counting the letters up to the end of the next word. Continue with new volunteers. Discuss the advantage of not having to count from the beginning every time.
Why keep track? Tell students you saw two students’ work. It looked like this: 3 8 11 14 20 T h e m o u s e a t e t h e a p p l e s. 3 8 11 13 19 T h e m o u s e a t e t h e a p p l e s.
ASK: Did the two students get different answers? (yes) What answers did they get? (19 and 20) Which answer is right? (20) Why? (we counted 20) Challenge students to find where the two students first got different numbers. ASK: Which word was counted incorrectly? (the second “the”) How does keeping track make it easy to see who is right? (it’s easier to see that the second student only counted 2 letters for “the” because 13 is only 2 more than 11; the number over “the” should be 3 more than 11). Emphasize that when you keep track, you can look for the first place the numbers start being different; that tells you which word was counted differently. Then re-count that word to see who is right.
Now write the following sentence on the board: A m o u s e r a n u p t h e c l o c k.
Have students write the number of letters after counting each word (1, 6, 9, 11, 14, 19) and compare their answers with a partner. ASK: Did you get the same final answer? Did you get the same numbers all the way through? If not, where do the numbers start to disagree? Can you tell who is correct? Give students lots of practice counting and keeping track and have them compare answers. EXAMPLES: Four dogs ran away. The leaves turned yellow. Today is Eric’s birthday.
Explain that this is called counting on. Write “counting on” on the board. SAY: Even expert mathematicians make mistakes with counting on. This is a way for them to check if they’ve counted correctly.
PROBLEM SOLVING
B-31
Number Sense 2-11
Have students complete BLM Count the Letters then exchange their BLMs with a partner. Did they get the same numbers all the way along in each sentence? Ask students to reflect on any mistakes. Did students make mistakes with longer words or shorter words? At the ends of sentences or at the beginnings? Closer to the end of the page than the beginning of the page?
PROBLEM SOLVING
B-32 Teacher’s Guide for Workbook 2.1
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CURRICULUM EXPECTATIONS Ontario: 1m25, 1m26; review, 2m3, 2m7 WNCP: 1N10; review, [R, C]
VOCABULARY numbers to 100 the reading pattern hundreds chart
Goals Students will use the reading pattern to count to 20 using a chart.
Count to 20 using a chart. Give students a long strip of thick paper with 20 squares labeled 1 through 20 and 20 two-coloured counters that fit on the squares. (If your counters are 2 cm wide, make the strip of paper 2 cm wide and 40 cm long.) Have students toss the counters and count the ones that turn up red by placing them on the chart in order, one counter per square. Repeat and have students record how many red counters come up each time.
The reading pattern. Write “cat” on the board and ask students what sound the “c” and the “t” make. Then say “cat.” SAY: Notice that you pronounce the “c” before the “t” (underline both letters). That’s because we read from left to right. Show left to right. Write the following sentence on chart paper, all on one line: “The cat sat on the red rug.” Ask students where the sentence starts and where the sentence ends. Then write “The big black cat sat on the” on one line and SAY: Oh, I’ve run out of paper. How can I finish writing the sentence? (start a new line) On the next line, write “small red rug.” Have students read the whole sentence together as a class. SAY: In English we read from left to right and from one line to the next line below; that’s our reading pattern. Write “reading pattern” on the board. Then write: “The big black cat sat on the small red rug and ate a grey round rat.” SAY: This sentence is very long and hard to read in one breath. Let’s divide the sentence into shorter lines. Show the line breaks in the margin.
ASK: Is this easier to read? Discuss how much easier it is to read the sentence this way. Write the sentence “His name / was Mark.” with the line break indicated. Give each student word cards for: his, name, was, Mark. Tell students to hold up the word they would read first, then the word that comes next, and so on to the end of the sentence. Ask students how they know which order to read the words in. Repeat for these sentences: “Was his / name Mark?” and “Mark was / his name.”
PRIOR KNOWLEDGE REQUIRED
Can count to 20 Can count to 20 using a chart
MATERIALS
a strip of 20 numbered squares and 20 two-coloured counters for each student (see below) 4 word cards for each student: Mark, was, his, name number cards for 1 through the total number of students in your class BLM 2-cm Grid Paper (p G-10)
PROBLEM SOLVING
Literacy
CONNECTION
The big black cat sat on the small red rug and ate a grey round rat.
B-33
Number Sense 2-12
The reading pattern with numbers. Remind students that counting with a chart from 1 to 10 was pretty easy (see NS2-4), but a chart marked 1 through 20 is harder to work with. ASK: How can we make it shorter and easier to work with? Does this problem remind you of another problem? How did we solve that problem? Help students make the connection to the reading pattern—you can break the long line into smaller lines. Students might suggest starting a new line at different numbers: 5, 10, 4, 6, or 7. Have various long sheets available to demonstrate all their suggestions.
Suggest that if they end the first line at 6, they make every line 6 squares long to make it look nicer. Then have students make their own chart for counting to 20 by cutting, arranging, and taping their long strip of numbered squares.
Use the reading pattern to find the next number. SAY: We read the numbers on a chart like we read text in a book: start at the left, go across the first row, then move to the next line and start at the left again. Because the numbers are not all on one line, it can be tricky to know where the next number is. For example, in the chart above, it’s not too hard to find 5 if I know where 4 is, but finding 7 is a bit harder. It’s not right beside 6 because we moved it. ASK: Where is 7? Can you find the number that comes right after 8, 12, 10, 18, 15, 16, and 19? Which numbers were harder to find: the big numbers or the small numbers? For which numbers was it harder to tell what comes next? (12 and 18) Why? (they are at the end of a row)
Have students copy charts onto BLM 2-cm Grid Paper. Teach them to do this accurately by counting the squares across and down. Alternatively, draw and photocopy charts for them.
The hundreds chart format. Draw the first two rows of a hundreds chart on the board. Discuss how this chart is different from or the same as the chart with rows of 6 or 7. ASK: How are the rows in this chart the same? (they are all the same length) Refer back to the 6 or 7 chart and ask if this was true there. (yes) Point out two numbers, one on top of the other, and shade them. ASK: What is the same about these numbers? (EXAMPLE: they both have 7’s) Is that the same for any number in the first row? If you look at the number below any number, do you see the same number with a one in front? (yes, except for 10) Refer back to the 6 or 7 chart and ask if this was true there. (no) Explain that rows of 10 are particularly useful because they are convenient for finding numbers. To find 17, look for 7 in the first row and then move down a row. Have students find: 9, 19, 4, 14, 3, 13, 15, 18, 11. Then have students find the numbers that come right after: 4, 14, 9,19, 17.
Count using the hundreds chart. Draw two rows of a hundreds chart and make 20 blank cards to fit. Give 16 cards to a volunteer to tape to the chart so they can be counted. Repeat with different numbers of cards and different volunteers. Emphasize the process for placing the cards: Start at 1; when you reach the end of a line go to the very beginning of the next line. To find how many squares are covered, students uncover the last number covered. Challenge: Predict how many squares are covered without uncovering the last square and then check the prediction.
PROBLEM SOLVING
PROBLEM SOLVING
EXAMPLE:
19 20
EXAMPLE:
Bonus 5 rows of 3 6 rows of 2 5 rows of 4 4 rows of 5
Extension—Use logical reasoning to guess numbers, including BLMs Guessing Numbers and BLM Guessing Number Game.
ONLINE GUIDE
Activity—students form a concrete number chart and perform the wave.
ONLINE GUIDE
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NS2-13 Adding Using a Chart Pages 20-23
CURRICULUM EXPECTATIONS Ontario: 1m25, 1m26; review, 2m1, 2m2, 2m3, 2m7 WNCP: 1N10; review, [R, C]
VOCABULARY reading pattern left right top bottom
Goals Students will add using cubes and then a chart.
Add using blocks. Give each student several red and blue 1 cm connecting cubes. Ask them to find 3 red blocks and 4 blue blocks. ASK: How many blocks is that altogether? Write on the board: 3 + 4 = 7. Repeat with various numbers of red and blue blocks, this time having volunteers write the addition sentence on the board.
Add using a chart and paper blocks. Draw the first two rows of a hundreds chart on the board, or use a large hundreds chart if available. Demonstrate how to find 3 + 4 by placing 3 red paper ones blocks and then 4 blue paper ones blocks on the chart in order, so that the last block is on square 7. Count as a class how many ones blocks there are altogether. Do several examples until someone notices that the last number with a block is always the total number of blocks. Then ask students to predict what the last block will be and check using several examples. Demonstrate putting 3 red and then 4 blue blocks on the chart randomly (not covering the first seven squares) and count them individually. Then put them on the chart in order, from 1, and count again. Ask students how the counting is already done for them when they put the blocks on in order. (Putting a card on the “1” is like holding it and saying “one”; the last card covered is like the last number said.)
Give each student a copy of BLM Hundreds Chart, ten red ones blocks, and ten blue ones blocks. Have students find 4 + 5 on their own hundreds charts. ASK: How is the adding done for you on the chart? Repeat using pairs of one-digit numbers that add to more than 10.
Use colouring and circling instead of blocks. Draw the first row of a hundreds chart on the board. Tell students that you want to add 3 + 5. Have a volunteer do so on the chart using the red and blue paper ones blocks.
PRIOR KNOWLEDGE REQUIRED
Can add Can read a hundreds chart Can count using a chart and otherwise
MATERIALS
a large hundreds chart and paper ones and blocks (red and blue) to fit BLM Hundreds Chart (p G-2) BLM Hundreds Chart — Three-Rows (p G-11) BLM Adding and Order (pp B-131–B-132) BLM Hundreds Chart — One-Row (p G-12) BLM Add Larger Numbers (p B-133)
NOTE: If you do not have red and blue ones blocks, you can use small connecting cubes, or else photocopy BLM Base Ten Materials onto red and blue paper.
PROBLEM SOLVING
PROBLEM SOLVING
Reflecting on what made the problem easy or hard, Making an organized list
TEACHING TIP: On Workbook pp. 21, 22 and the BLMs, some students may need to do each step separately; do the first step for all questions first, then go back and do the second step.
B-35
Number Sense 2-13
SAY: Now let’s try something different: instead of putting on 3 red paper blocks, let’s just shade the first 3 squares. (remove blocks and shade the 3 squares) Also, instead of putting on the 5 blue paper blocks, let’s just circle the next numbers. (remove blocks and circle the next 5 squares) We can now see that 3 + 5 = 8 since 8 is the last number circled. ASK: Do you think this way is quicker and easier than using blocks? Discuss.
Practice. Draw the first two rows of a hundreds chart on the board. Use it to add pairs of one-digit numb