Grade 1: Unitizing and Skip-counting by 2s, 5s, and 10s Multiplication and Division Horizontal Content Strand Texas Essential Knowledge and Skills (TEKS): (1.1) Number, operation, and quantitative reasoning. The student uses whole numbers to describe and compare quantities. The student is expected to: (A) Compare and order whole numbers up to 99 (less than, greater than, or equal to) using sets of concrete objects and pictorial models; (B) Create sets of tens and ones using concrete objects to describe, compare, and order whole numbers; (C) Read and write numbers to 99 to describe sets of concrete objects. (1.4) Patterns, relationships, and algebraic thinking. The student uses repeating patterns and additive patterns to make predictions. The student is expected to identify, describe, and extend concrete and pictorial patterns in order to make predictions and solve problems (1.5) Patterns, relationships, and algebraic thinking. The student recognizes patterns in numbers and operations. The student is expected to: (A) Use patterns to skip count by twos, fives, and tens; (B) Find patterns in numbers, including odd and even; (1.7) Measurement. The student directly compares the attributes of length, area, weight/mass, capacity, and temperature. The student uses comparative language to solve problems and answer questions. The student selects and uses nonstandard units to describe length. The student is expected to: (A) Estimate and measure length using nonstandard units such as paper clips or sides of color tiles; (B) Compare and order two or more concrete objects according to length (from longest to shortest); (1.11) Underlying Processes and Tools. The student applies Grade 1 mathematics to solve problems connected to everyday experiences and activities in and outside of school. The student is expected to: (B) Solve problems with guidance that incorporates the processes of understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness; (C) Select or develop an appropriate problem-solving plan or strategy including drawing a picture, looking for a pattern, systematic guessing and checking, or acting it out in order to solve a problem; (D) Use tools such as real objects, manipulatives, and technology to solve problems. TEKS Connections to Other Grade Level Strands: Number and Operations (Multiplication and Division) Algebra (Patterns) Measurement Mathematics TEKS Connections: Grades K-2 380
Purpose: This lesson introduces first graders to two important knowledge and skills skip-counting and unitizing. By internalizing these concepts, children can begin to apply more efficient quantification strategies involving repeated addition and eventually formal multiplication for solving word problems. The unitizing nature of skip-counting lays the foundation for multiplication by helping a child to organize a set of objects into clusters of equal groups, thereby allowing him or her to land predictably on designated numbers and pass over others. By the end of first grade, children should be able to use skip-counting beyond a mere rote level, incorporating it as a strategy, in addition to one-to-one correspondence, to quantify rationally large sets of objects. Suggested Vocabulary: equal groups measure skip-counting length repeated addition total Materials: Baggies each containing all ten of the basic proportional fraction manipulatives such as Cuisenaire rods - per pairs of students Chart Paper Linking Cubes such as Unifix cubes Class-size 100s Chart with transparent squares for marking numbers Markers and paper per student Advanced Preparation: Handout 1-2: Riddle Cards For Each Student: Markers and paper Handout 1-3: Riddle Recording Sheet Handout 1-1: Individual student 100s Chart For Each Pair of Students: Baggies each containing all ten of the basic proportional fraction manipulatives such as Cuisenaire rods Suggested Pacing: 3 days Mathematics TEKS Connections: Grades K-2 381
Procedures Engage: 1. Distribute to pairs of students baggies containing a set of each of the ten different proportional fraction manipulatives such as Cuisenaire rods. These bars are called Cuisenaire rods. I am going to let you explore them with your partner for about 5 minutes. Study them carefully and be ready to share your discoveries to the whole group. 2. As the students explore, observe and listen to their conversations. Pay attention to students who try to assign numerical values to each rod. Make note of the strategies these students use to figure out what each rod is worth. 3. After the five minutes of free exploration, call on volunteers to share their observations and discoveries. If the rods are new to students, this exploration will take more than five minutes. 4. If during this initial discussion the students fail to explain or express inquiries on how to figure out the value of each rod, guide the class in thinking about these ideas through the following questions: How could you order these rods from shortest to longest/ tallest? include: White is the shortest, followed by red, then light green, etc. Does the way you order the rods by length help you figure out the numerical value of each rod? include: If there are ten rods, then white is equal to 1, red is equal to 2, etc. Teacher Notes Materials There are ten basic bar units within a set of proportional fraction manipulatives such as Cuisenaire rods. They are, from shortest to longest: 1. White (Tan) 2. Red 3. Light green 4. Violet 5. Yellow 6. Dark green 7. Black 8. Brown 9. Blue 10. Orange Informal Assessment As students are exploring (Refer to Step 2), look for the following attempts initiated by students to figure out of the size and numerical value of each rod: Do the students order the rods from shortest to longest and then iterate a shorter bar against a longer bar to figure out how many of the former will equal the length of the latter? TEKS Connection Unit iteration is a measurement concept referred to in TEKS 1.7A Do students immediately figure out that the shortest rod is worth one unit by comparing it with the other 10 rods? Do they then determine that the numerical value of each bar must correspond to its ordinal position relative to its size when the rods are ordered from shortest to longest? If after figuring out the value of shorter rods, do students relate this understanding and apply number facts (e.g., 2 + 2 = 4, so 2 reds = 1 purple) to determine the value of longer rods? Mathematics TEKS Connections: Grades K-2 382
If you claim that the longest rod (the orange one) is worth 10, could you use any of the other rods to help you prove your statement? include: I knew just by looking [comparing] that it takes 2 white rods to make one red. Then, I saw that the red rod is half of the purple rod, so I knew that 2+2 = 4. Then, I put a white rod (1) with a purple rod (4) and saw that it was the same size as a yellow rod. So then I knew that a yellow rod is equal to 5 because I know that 1+ 4 = 5. Then, I saw that a yellow rod is half of an orange rod. So I knew that an orange rod is worth 10 because 5 + 5 = 10. 5. Explain to the class that they will now measure how long objects in the classroom are using the red rod as a non-standard unit. Use this scenario to set the stage for solving the following word problem: Since there are not enough red rods for every student to measure with during tomorrow s math lesson, you will each make your own red rods using interlocking cubes. Since a red rod is worth two units, you will need two white interlocking cubes for every rod you make. If each student is supposed to make 9 red rods, how many white interlocking cubes will you need? Responses may vary. Strategy for ALL Learners So that all students are able to make sense of the strategies explained by classmates during the sharing session outlined in Step 4, illustrate and label each student s response on chart paper. Figure A Jessica s strategy: 1+1 = 2 + = 2+2 = 4 + = 1+4 = 5 + = So, 5+5 = 10 6. Remind the class that they must first solve the problem described in Step 5 then record their strategy on paper using numbers, pictures, or words. Problem-Solving Strategy To ensure that the students make sense of and comprehend the word problem described in Step #5, call on various students to summarize in their own words what you said. Mathematics TEKS Connections: Grades K-2 383
Explore: 1. Distribute sets of interlocking cubes to pairs of students. Every child will need markers and a sheet of paper for recording his or her work. 2. As children are working, move about from student to student, asking questions to probe for understanding, redirecting misconceptions, and taking note of different problem-solving strategies. If students are having difficulty getting started, ask the following questions: When you make one red rod, how many cubes do you have? Two When you make two red rods, how many cubes do you have? Four What about when you have three rods? I will have six cubes. What is happening to the amount each time you make more rods? Do you notice a pattern? include: It is getting bigger, or it keeps going up two more each time I make one more. 3. When students finish, ask them to demonstrate how they figured out how many cubes they would need to make 9 red rod units. Make note of 3 different problem solving levels. (For a description of these levels, refer to the teacher notes.) 4. Allow students who finish sooner than others to connect the red rods they made into one train of 18 connecting cubes. Then, have the children make size comparisons with their train, determining which objects are about as long, longer than, or shorter than a series of 9 red rod cube units. Instruct students to record their findings on a sheet of paper. Research Fennema, Carpenter, Levi, Franke, & Empson (1999) have conducted considerable research on the informal strategies children use to solve word problems. Below are three different levels of quantification strategies that Fennema, et al., have observed across several student populations: (The examples below relate to strategies for solving multiplicative word problems.) Direct Modeling: The student models the structure of the problem, making stacks of each group with counters, then counts all of the objects one-by-one in every group to figure out the total. Counting Strategies: Instead of counting each object one-by-one, the student skip counts, enumerating the accumulating total or each group to determine how many there are altogether. Numerical Reasoning: The student uses derived number facts or repeated addition (e.g., I know that 2 + 2 = 4 and then I made another group of 4 so that I could add 4 + 4 to get 8. I counted 10 over here, and so 10 + 8 = 18. There are 18 altogether. ) Mathematics TEKS Connections: Grades K-2 384
Explain: 1. Gather the class together to debrief students problem-solving strategies and discoveries. 2. Select students to share who solved the problem using one of the quantification strategies outlined in the teacher notes from the Explore section. 3. As these students share, summarize and label their strategies on chart paper. Make sure that each selected student demonstrates his or her counting strategy. See the Pacing explanation in the teacher notes column. 4. As a whole group, compare and contrast the different strategies that were shared: How are these different problem-solving solutions alike? include: Everyone said they needed 18 cubes to make 9 red rods or they all made groups. How is each of these students approach different from the other? Did everyone use counting to figure out the total? include: [Student s name] counted all of his cubes, but [Student s name] only counted the last numbers in each rod. 5. Discuss key mathematical vocabulary. For example, if some students point out that they added in order to figure out the total, introduce the terms repeated addition and equal groups. Lead the students to this understanding through questioning: How is adding together one group of 2, another group of 2, and another group of 2 different from adding a group of 4, 5, and 3? include: In the first type of adding, all of the groups are equal, in the second example, all of the groups have different numbers. Then, make the targeted vocabulary explicit: When you join together equal groups, you are using repeated addition. Scheduling Make sure that the students have been physically active before the sharing session so that they are ready to concentrate and listen to classmates strategies. Pacing In order to go more in depth into key mathematical ideas, select only 3 different strategies to share. Allow those students who were not selected to share to vote on which of the 3 strategies is closest to the approach that they used. (See Figure B below:) Whose strategy is most like the one you used to solve the How many cubes to make 9 red rods problem? Stephen s Counting all Lisa Tyrone Moesha Tonya s Skip-counting Denise Lyle Sunny Harold Lupe s Number facts Anthony Ariel Vocabulary To make the concept of repeated addition more concrete, relate it to students background and familiarity with repeating patterns ABB ABB ABB. Point out how the unit of the pattern (the part of the arrangement that repeats) continues over and over, similar to the situation modeled in the word problem in which equal sets of 2 were repeatedly joined together. Materials A 100s Chart is a powerful visual model for illustrating counting patterns and representing place value. As children go from left to right horizontally on the chart, they see the repetitive cycle of the digits 1 9 in the ones place. This same pattern emerges in the tens place, moving vertically from top to bottom. Mathematics TEKS Connections: Grades K-2 385
Elaborate: 1. Display a class size 100s Chart. Connect the skipcounting and grouping that some of the students demonstrated in the Explain section to how the 100s chart can be used as a tool to visualize and keep track of this type of counting and quantifying. 2. Begin by placing a transparent marker or square over the numbers on the chart where each of the nine red rods would stop. Engage the students in this activity by asking them which numbers you should cover: On which number do I land when I count the first rod? 2 The second rod? 4 The third rod? 6 etc. What do you notice about the 100s chart when I place a marker over every number we land on after counting each of the rods? include: It makes an AB pattern, or it skips every other number, or it lands on all of the even numbers. Research Clements, Samara, and DiBiase (2002) site evidence gathered by a panel of researchers indicating that most typically developing 6- and 7- year-olds are just learning to skip count meaningfully when quantifying sets of objects and can begin to do with groupings of 2, 5, and 10. By age 7 and 8, children begin to develop greater facility skip-counting by less regular numbers, such as 3 and 4. 3. Begin to introduce other skip-counting rules, such as counting by 5s and 10s. Continue to use the rods, connecting cubes, and the 100s chart to illustrate these counting methods. For example, show a yellow rod and an orange rod. What is the value of the yellow rod? 5 The orange rod? 10 How many squares would I take up on the 100s chart to represent 3 yellow rods? 15 4. Call on a student volunteer to demonstrate how to count out 5 yellow connecting cubes for each of the 3 yellow rods. Then, count and mark every 5 th numeral on the 100s chart to show a counting by 5s pattern. Follow the same steps to illustrate skipcounting by 10s using the orange rod. Mathematics TEKS Connections: Grades K-2 386
5. Discuss the patterns that unfold on the 100s chart as you mark the numbers the students land on when they count by 5s and 10s. (See Figure C in the teacher notes section.) Figure C Evaluate: 1. Display the stack of Riddle cards (Refer to Handout 1-3: Riddle). Read Card A: I am made up of 5 yellow Cuisenaire rods. How many cubes do Handout 1-3: Riddle 2. Gather suggestions from students about how to solve the riddle. Remind them of the work they did during the Explore section solving the 9 red rods problem. 3. Demonstrate how to record your counting on an individual 100s chart Handout 1-1: Student 100s Chart Recording Sheet by encircling groups of 5, starting at 1, then continuing until there are 5 groups of 5 squares circled, ending at 25. 4. Model how to record the number of rods, the number of cubes needed to make each rod, and the total number of cubes in the indicated columns on Handout 1-3: Riddle. Pacing The activity outlined in the Evaluate section would work best as a follow-up lesson scheduled during a different block of time or, preferably, on the following school day. 5. Distribute the Student 100s Chart and Riddle handout. Students will also need sets of connecting cubes. 6. As a formal assessment, have the students solve their riddles independently. Mathematics TEKS Connections: Grades K-2 387
7. Differentiate as needed based on your observations of the students counting. Refer to the Teacher Notes section for ideas on how to incorporate appropriate scaffolding. 8. When students finish, have them put together their rods into one train and use that train to measure objects in the classroom that are the same length. Scaffolding The Riddle Cards are differentiated to accommodate children s counting range. Select a card that will meet the instructional level of each student. Based on the students performance, try the following interventions and challenges: Below: If some children are having difficulty keeping track of the number of groups they need to make, in addition to the number of cubes they need to count into each group, provide 5 or 10 frames to help these students organize their count. Developing: If some children are still relying on one-to-one correspondence to count all of the cubes, point to and recite the last number of each group encircled on the 100s chart while the student attempts to correspond his or her enumeration of the set of objects to your lead. Achieving: If some children are skip-counting with ease and flexibility, provide these students with an extension activity using a calculator. Have them program a skip-counting command e.g., [+2], [+5], [+10] into the calculator and then continue pressing the [=] button until they reach their target number. Mathematics TEKS Connections: Grades K-2 388
Handout 1-1: Individual Student 100s Chart s 100 s chart 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 9 98 99 100 Mathematics TEKS Connections: Grades K-2 389
Handout 1-2: Riddle Cards I am made up of 5 yellow (Card A) I am made up of 3 red rods. (Card B) I am made up of 4 orange (Card C) I am made up of 4 yellow (Card D) I am made up of 11 red rods. (Card E) I am made up of 6 orange (Card F) I am made up of 7 yellow (Card G) I am made up of 5 red rods. (Card H) Mathematics TEKS Connections: Grades K-2 390
I am made up of 9 orange (Card I) I am made up of 8 yellow (Card J) I am made up of 6 red rods. (Card K) I am made up of 8 orange (Card L) I am made up of 4 yellow (Card M) I am made up of 12 red rods. (Card N) I am made up of 5 orange (Card O) I am made up of 10 yellow (Card P) I am made up of 7 red rods. (Card Q) I am made up of 4 orange (Card R) Mathematics TEKS Connections: Grades K-2 391
I am made up of 6 yellow (Card S) I am made up of 4 red rods. (Card T) I am made up of 7 orange (Card U) I am made up of 9 yellow (Card V) I am made up of 8 red rods. (Card W) I am made up of 2 orange (Card X) I am made up of 2 yellow (Card Y) I am made up of 20 red rods. (Card Z) Mathematics TEKS Connections: Grades K-2 392
Handout 1-3: Riddle Name: Card #: Riddle: Color of Rod Number of Rods Total Cubes Number of Cubes: Show how you solved the riddle and counted the cubes. Use numbers, pictures, and words. Mathematics TEKS Connections: Grades K-2 393