Beyond Counting by Ones

Beyond Counting by Ones Mathematical Activities for Developing Number Sense and Reasoning in Young Children Dr. DeAnn Huinker University of Wisconsin-Milwaukee February 2000 DeAnn Huinker, University of Wisconsin-Milwaukee (2/22/2000) 1

Thinking in Groups Seeing twoness and threeness Hide three pennies. Uncover the pennies and ask, How many pennies are there? Watch to see if the child counts by ones, or sees the group as a whole. Repeat with two pennies. Encourage a quick response. The goal is to tell how many in a one-second glance. Making twos and threes The objective is to see and make twos, not to see how many twos. Give each child 10 counters. Have them break the set into parts by making twos. Then put the whole set of objects back together and repeat until the twos are made quickly. Next start with 12 counters and make sets of three. Initially it may take a minute, but with practice children can accomplish these tasks in almost 10 seconds. Dot Cubes Dot cubes with only one, two, or three dots on the sides are useful for young children. You can draw dots on blank wooden cubes or cover regular dot cubes with stick on labels. Once children are able to subitize these quantities then begin using dot cubes with one through six dots on the sides. As children roll the cubes, often ask, How many dots are there? and How did you see them? Two children can play, Who has more dots? or Who has fewer dots? by each rolling one cube and then comparing the number of dots. Observe the strategies used by the children to compare. Dominoes Have children to explore with dominoes. Ask them to describe the various arrangements of dots, How are the dots arranged? What do you notice about the dots? The dominoes can be used for various activities, such as finding all dominoes with a total of six dots, seeing how quickly one can tell how many dots are on each side of a domino, finding the doubles, or making matches of adjacent dominoes. Observe children to see if they count by ones, use groupings, or are able to identify the quantities. Dot Plates and Dot Cards Prepare a set of paper plates, cards, or transparencies with groupings of two through ten dots in many different arrangements. Use the dot plates/cards for various activities as described on the Dot Pattern Activities sheet. Often ask children, How many dots do you see? and How did you see them? DeAnn Huinker, University of Wisconsin-Milwaukee (2/22/2000) 2

Dot Pattern Activities Dot plates or dot cards consist of patterned sets using dots. The dot patterns represent quantities from zero through ten in different configurations. The patterns show the common dice and domino patterns, as well as many other variations that are combinations of two smaller patterns or a pattern with one or two additional dots. Some of the patterns can also be made with two colors to more clearly show the two parts. Keep the patterns compact so they are easier to see. The dots plates or cards can easily be made with paper plates and stick-on-dots. Flash Math Hold up a dot pattern for about 3 seconds. The children should visualize the dots and try to figure out how many dots were on in the pattern. Show the plate again for another 3 seconds. Then ask, "How many dots did you see?" The students can respond in unison. Set up some type of signal so that children do not call out the number of dots until all everyone has had a chance to visualize and think about the pattern. Then ask a student to explain, "How did you see it?" Have several students explain how they saw the dots. There is no right or wrong way to see a pattern and often students see them in different ways. Begin with easy patterns and then add a few new patterns each day. Make the Pattern Flash a dot pattern for a few seconds. Then tell the students to make the pattern they saw using counters on their work mats. It is helpful to have a work mat so that you can easily see the pattern made by the student. Spend some time discussing the configuration of the pattern and the number of dots. Hold Up Flash a dot pattern. Then tell students to hold up that many fingers. Next discuss how the students saw the dots, e.g. as a three and a two or as a four and a one. Show the Numeral Flash a dot plate. Tell the students to hold up the numeral card that tells how many dots. Then discuss the configuration of the pattern. One More Than or One Less Than Flash a dot pattern and have the students tell the number that is one more than (or one less than) the number of dots. Have the students explain their reasoning. Write the equation Flash a dot pattern. Each student writes an equation that reflects how they show the dots. Compare Working in pairs, each student displays a dot pattern. Then they discuss, "Which plate has more dots?" Combine Working in pairs, each student displays a dot pattern. They they figure out, "How many dots all together on both cards (plates)?" DeAnn Huinker, University of Wisconsin-Milwaukee (2/22/2000) 3

Dot Patterns DeAnn Huinker, University of Wisconsin-Milwaukee (2/22/2000) 4

Part-Whole Activities Part-whole mats Children can place a pencil or piece of string down the middle of a work mat or be given a mat that clearly shows two parts. Tell the students to place a specific number of counters, e.g. six, on their mat. They should place part of the counters on the left side and part of the counters on the right side. This leads to an interesting discussion as to whether zero is considered a part of the number. This is something your class can decide. The children can be asked to represent their combinations in some way on paper. The different combinations can also be recorded in a table on the board for discussion. Story Boards Children can design their own story boards, such as an ocean scene, school yard, playground, or any place that is special to them. Then using a designated number, such as six, of counters, teddy bears, fish, or other shapes, have the children tell part-whole stories. For example, There are five fish swimming in the ocean. Three of the fish are swimming at the top of the ocean and two fish are swimming at the bottom of the ocean. The stories can be recorded in a whole class journal, dictated to older students or an adult to be recorded, or just shared orally. Shake and Spill Tell the students to place a specific number of two color counters, e.g. five, in a cup or in their hands. Next they shake up the counters and then spill them onto a work mat (felt works well) on their desks. Children report how many counters are red and how many are yellow. The different combinations can be recorded in a table for discussion. Children can also fold a paper into fourths and record a combination from each toss in each section. These can be cut apart and graphed. This activity can lead to an interesting discussion on the likelihood of certain combinations, as well as to a discussion on all the possible combinations. How Many of Each Problems Seven Peas and Carrots Tell students that you have seven things on your plate. Some of them are peas and some of them are carrots. Explain that there is more than one correct solution to the problem. The students may use counters to help them solve the problem. When students have found a solution, they record it on blank paper using pictures, numbers, and/or words. The number of objects and the kinds of objects can be varied as this type of problem is revisited throughout the school year. Source #1: Schulman, L, & Eston, R. (1998, October). A problem worth revisiting, Teaching Children Mathematics, 5, 72-77. Source #2: Kilman, M., Russell, S. J., Wright, T., & Mokros, J. (1998). Mathematical thinking at grade 1. White Plains, NY: Dale Seymour Publications, p. 43-53. Problems About 10 Have children make up their own problem about ten things. The problem should be about two different kinds of things, such as apples and bananas or turtles and rabbits. The children first select the two kinds of things. Then they decide how many of each there are to have ten things all together. The children may use counters or other materials to help them. Then they record their problem on paper. Source: Kilman, M., & Russell, S. J. (1998). Building number sense. White Plains, NY: Dale Seymour Publications, p. 48-52. DeAnn Huinker, University of Wisconsin-Milwaukee (2/22/2000) 5

Towers of Six (Seven, Eight, Nine, or Ten) Working in pairs, children build three towers of six together. Each child uses a different color of interlocking cubes, e.g. red and brown. The children take turns rolling two dot cubes with one, two, and three dots on the sides. (If building larger towers, use regular dot cubes.) If the first child rolls a two and a two, she would use the red cubes to build a tower four cubes high. Then the second child rolls the cubes and gets, for example, a three and a two. This child would add two brown cubes to make a complete tower of six and then start a new tower with the remaining three cubes. They continue to play until they have three complete towers. It is not necessary to roll an exact number to finish the game. The children then find the number of cubes of each color in each tower and record this information on paper using pictures, numbers, and/or words. Source: Kilman, M., Mainhart, Murray, M., & Economopoulo, K. (1998). How many in all? White Plains, NY: Dale Seymour Publications, p. 42-44. Hiding Hands Tell the students to place a specific number of counters, e.g. five, into their hands. Then they should put their hands with the counters under their desks and secretly put part of the counters in one hand and the other part of the counters in the other hand. Now the children take turns revealing their hidden parts by first showing the counters in one hand and then showing the counters in the other hand. For example, Morgan has three counters and two counters. When children become familiar with the combinations, ask for predictions before the counters in the second part or hand is revealed. Observe which students need to see the counters and which do not. In either case, what strategies do children us to determine the hidden part. Do they know the combination? Do they count by ones? Do they use grouping strategies? Do they visualize the hidden part? Under the Cup In pairs, children take turns hiding counters in a cup. Assign each pair a number to work on or let them select a number. One child secretly hides part of the counters under the cup and places the remaining counters on the table. The other child figures out how many counters are hidden and then shares his/her strategy. The number of counters in each part is recorded. Then the children continue taking turns as they repeat the activity with the same amount of counters. When they have found many different combinations, they can select a new number of counters and begin again. Turn over Five The object of this Concentration-type game is to find pairs of cards that add to 5. Children play in pairs or small groups. Use a set of twenty-four numeral cards with four cards for each of the numbers 0 through 5. Lay the cards facedown in four rows of six. Players take turns choosing two cards to turn over, trying to find a combination that adds to 5. The game can be varied by selecting a different target sum, such as 10. Source: Kline, K. (1999, April). Helping at home. Teaching Children Mathematics, 5(8), 456-460. Fives Go Fish This game is a variation of the popular game of go fish. The same rules apply, but rather than search for pairs of the same number, a player searches for pairs of two numbers that add to 5 or to other numbers as appropriate. Use a set of twenty-four numeral cards with four cards for each of the numbers 0 through 5. Deal out six cards to each child. Place the rest of the cards face down in the pond or center of the group. Children take turns asking each other for a specific card, such as "Mary, do you have a three?" If Mary has a three she must give it to the person asking. If she does not have a three she tells the person to "Go Fish" and the person draws one card from the pond. Source: Kline, K. (1999, April). Helping at home. Teaching Children Mathematics, 5(8), 456-460. DeAnn Huinker, University of Wisconsin-Milwaukee (2/22/2000) 6

Ten Frame Activities Numbers are represented on a ten-frame by filling in, from left to right, the top row and then the bottom row. Each number is then seen simultaneously in relationship to five and ten. Fill Ten frames Call out numbers between one and ten in random order and have children represent the numbers on their ten-frame. Discuss the relationships to five and ten. Flash Ten frames Flash ten-frame cards to the class to see how fast they can tell how many dots are shown. Frequently spend three to five minutes with this activity. Encourage children to see how quickly they can name the amount. An alternative to saying the number of dots is to say the number of empty spaces. Ten-frame Facts Flash ten-frame cards to the class. The children should state the ten-fact, for example if seven dots are shown, they state, "Seven and three." A variation is to have the children write the equation, 7 + 3. Five And Draw a large ten-frame on the chalkboard or use an overhead transparency. Encourage the children to think in terms of the ten-frame. Call out a number between five and ten. Children respond with "five and " using the appropriate number. For example, the teacher calls "Eight" and the children respond, "Five and three." Make Ten Show a ten-frame. Encourage the children to think in terms of the ten-frame. The teacher calls out a number between zero and nine. The children respond with the number required to make ten. For example, if you call "Four," the children respond with "Six." One or Two More, or Less Flash ten-frame cards to the class and have the children respond with the number that is one more than the number shown. When the children are good at this, have them respond with the number that is two more. For even more of a challenge, have the children respond with the number that is one or two less than the number shown. Crazy Mixed-Up Numbers Each child has a ten-frame and counters. The teacher calls out numbers from zero to ten. The children figure out what changes are needed to make the new number. The children say "plus " or "minus " [whatever is required to change their ten-frame to the new number.] They then add or remove counters accordingly. Small groups of children can use long strips of paper with a list of about 15 "crazy mixed-up numbers." A student can read one number at a time to the rest of the players. DeAnn Huinker, University of Wisconsin-Milwaukee (2/22/2000) 7

Collect Ten Children play in pairs or small groups. Each person has a ten frame and ten counters. Children take turns rolling a dot cube that has one, two, or three dots on a side. If a three is rolled, the child places that many counters onto his/her ten frame. The goal is to fill the ten frame. The exact amount must be rolled. If the number is too big, then the child must try again on his/her next turn. Fill-Up Together Two or three children play as a group. Place two or three ten frames in the middle of the group. Each person needs 10 to 20 counters. The goal is for the group to fill all the ten frames by working together. working together. The players take turns rolling a dot cube and placing that many counters on a ten frame. Each ten frame must be filled from left to right, top row then bottom row. A number may not be split up. If the player is unable to fit the entire number in a ten frame, they must start a new ten frame. If this is not possible the player will lose a turn. The game is over when all the ten frames are filled. Cover Up The object of this game is to cover up two 10-frames completely using interlocking cubes. To prepare for the game, make eight trains of one color that are five cubes long and twenty trains of another color that are two cubes long. Also have about forty single cubes of yet another color available. The player rolls a number cube and takes that many cubes to cover the 10-frames on their game board. The player may take any combination of cubes to make the total rolled but is not allowed to pull apart any of the 2- trains or 5-trains. For example, a player who rolls a 6 may take one 5-train and one single cube or may take two 2-trains and two single cubes. Play continues until one player has completely covered up both 10-frames with interlocking cubes. Source: Kline, K. (1999, April). Helping at home. Teaching Children Mathematics, 5(8), 456-460. DeAnn Huinker, University of Wisconsin-Milwaukee (2/22/2000) 8

Counting Calculator Activities for Developing Number Relationships Count by ones. 0 + 1 = = = = Count backwards by ones. 10 1 = = = = Skip count: Count by fives. 0 + 5 = = = = Count by tens. 0 + 10 = = = = Counting-on by ones. 5 + 1 = = = Counting-on by tens. 23 + 10 = = = = Number Relationships One More Than Machine Program the calculator: 0 + 1 = Use it: 3 = 10 = 8 = 32 = Two More Than Machine Program the calculator: 0 + 2 = Use it: 7 = 5 = 10 = 23 = One Less Than Machine Program the calculator: 1 1 = Use it: 9 = 20 = 5 = 48 = Two Less Than Machine Program the calculator: 2 2 = Use it: 6 = 10 = 5 = 9 = Part-Whole Machine Program the calculator: 10 10 = Use it: 3 = 4 = 8 = 7 = (Ignore the negative sign. ) Doubler Machine Program the calculator: 2 x 1 = Use it: 6 = 4 = 11 = 25 = DeAnn Huinker, University of Wisconsin-Milwaukee (2/22/2000) 9

Basic Fact Practice Practice adding seven. Program the calculator: 0 + 7 = Use it: 2 = 10 = 8 = 32 = Practice subtracting three. Program the calculator: 3 3 = Use it: 9 = 15 = 7 = 28 = Practice multiplying by nine. Program the calculator: 9 x 1 = Use it: 3 = 5 = 10 = 7 = Practice dividing by five. Program the calculator: 5 5 = Use it: 40 = 15 = 100 = 37 = Place Value Activities Wipe Out Enter a number into the calculator, for example 458. Challenge your opponent to wipe out the 5 without changing any of the other digits. Then challenge your opponent to wipe out the 7 and then the 4. A point can be awarded for each correct move. Change it Enter a number into the calculator, for example 28. Challenge your opponent to change the 2 to a 5 without changing the other digit. Then challenge your opponent to change the 8 to a 6 without changing the other digit. A point can be awarded for each correct move. This game can go on forever. Make a New Number Enter a number into the calculator, for example 359. Challenge students to make new numbers without pressing the clear key. For example, change 359 to 659 to 679 to 609 to 209 to 239 and finally to 235 For each change, students should record the keys they pressed to modify the number. For example, to change 359 to 659, students might record + 300 or + 100 + 100 + 100. Silent Races Select a number for skip counting, such as 2, 5, 10, or 25. Then select a target: e.g. 50, 100, 75, or 300. Each student then programs his or her calculator: e.g. 0 + 5 = On the teacher s mark, the students begin the skip counting race by repeatedly pushing the equal key on their calculators. When a student reaches the target number, he or she stands up. If a student goes past the target number, he or she must start over. DeAnn Huinker, University of Wisconsin-Milwaukee (2/22/2000) 10