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32 Josie Number Corner Student Book 2 Quarts or Spill, Game page of 2 Blue /22 NAME TEAM COLOR DATE COMPUTATIONAL FLUENCY Spin (cups) Convert to Ounces Running Total (oz) Quarts & Cups c 8 oz 8 oz cup 2 c 2 oz 20 oz 2 cups c 2 oz 32 oz quart c 4 oz 36 oz qt cup c 2 oz 48 oz qt 2 cups (or qt) c 8 oz 56 oz qt 3 cups First Quart Second Quart Calendar Collector November Sunday Monday Tuesday Wednesday Thursday Friday Saturday Calendar Grid M arker & Date Equivalent Expressions M oney F raction s Decimal s dime cents ( ) It s worth pennies or 2 nickels. 20 of a dollar of a dollar. $0. row on the mat is kind of like dime. They re both of the whole unit (ten hundreths). (one tenth) Each square is like a dime because there are of them in the whole rectangle. of a rectangle. (one tenth) 2 dimes 20 4 nickels 20 0 or If the square was a dollar, the 2 rows could be 2 dimes or It s and of a dollar is 20. or if you split each piece in half..2 It s like 2 dimes..2 quarter 25 2 dimes and a nickel $ Number Corner, Grade 5 2

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34 November November Calendar Grid CALENDAR GRID Fractions & Decimals Overview The pattern this month features fractions and decimals tenths, fifths, fourths, and halves shown as parts of a dollar, a -by- grid (base ten mat), a rectangle divided into parts, and a pentagon divided into 5 parts. The workouts that go with this pattern introduce concepts about fractions and decimals that will be developed in greater depth later in the school year during Number Corner and your regular math program. Frequency Update the Calendar Grid each day and share observations and predictions once or twice a week. Update the record sheet every day after the second workout. Skills & Concepts H ordering, modeling, and comparing fractions and decimals H demonstrating meanings for fractions in different contexts H recognizing equivalent forms of common fractions and decimals H adding fractions You ll need H November Miniature Markers 3 (Overhead NC 3.3) H Calendar Grid Units (Overhead NC 3.4) H Calendar Grid Units (Blackline NC 3., copy posted beside the Calendar Grid before the first workout) H Recording Chart Panels (Blacklines NC 3.2 and 3.3, see Advance Preparation) H November Mini-Markers, pages 3 (Blackines NC , copy, see Advance Preparation) H Base Ten Mats (Number Corner Student Book, page 39) H Thinking About Fractions & Decimals (Number Corner Student Book, page 53) H More About Fractions & Decimals (optional, Number Corner Student Book, page 58) H Calendar Grid pocket chart H Day, Month, and Year markers H Fraction and Decimal calendar markers H overhead coins if you have them H overhead pens H piece of paper to mask portions of the overhead Number Corner, Grade 5 35

35 November Calendar Grid Fractions & Decimals (cont.) Advance Preparation Use copy of the first Recording Chart Panel (Blackline NC 3.2) and copies of the second panel (Blackline NC 3.3) to prepare a recording chart for use during and after the second Calendar Grid Workout. You may want to laminate this chart so you can reuse it in future years. M arker & Date Equivalent Expressions Money Fractions Decimals Run a single copy each of Blacklines Cut the Mini-Markers apart and store them in an envelope where you and your calendar helpers can find them easily. Week Introducing and Discussing Markers 3 Open the first Calendar Grid Workout by displaying the top portion of November Miniature Markers 3 at the overhead. Give students a minute to examine the information quietly. Next, ask them to pair-share any mathematical observations they can make. Then invite volunteers to share comments and questions with the class. November Overhead NC 3.3 November Miniature Markers 3 = unit = unit = unit = unit 36 Number Corner, Grade 5

36 Calendar Grid Fractions & Decimals (cont.) November Students It says they re all worth, but they re different shapes and sizes. Yeah, like that top one is a -by- square and then there s a dollar bill. And the rectangle is made out of squares, and the pentagon is divided into 5 parts. Teacher How can these different things all have a value of unit? Students Maybe you can say that two things are both worth, even though they re different sizes. Like a regular candy bar and a king-size one. The king-size bar is bigger. They re both worth, but I d rather have half of the bigger one! I have an idea about the square and the dollar. There are 0 little squares in the big square, and 0 pennies in a dollar, so they re kind of alike even though they re different sizes. Expose the first calendar marker on the overhead and ask students to suggest different ways to name what they see. Record their ideas in the appropriate boxes, taking care not to show the rest of the overhead for now. November Overhead NC 3.3 November Miniature Markers 3 = unit = unit = unit = unit Equivalent Expressions Marker Money Fractions Decimals dime cents ( ) It s worth pennies or 2 nickels. row on the mat of a dollar. $0.. (ten When you have recorded students ideas, reveal the second marker on the overhead and ask students to turn to Base Ten Mats on page 39 in their Number Corner Student Books. Ask them to color in the first mat to match what is shown on the overhead and record two or three different expressions to label the amount they have colored in. Give them some time to compare ideas with their neighbors and then ask volunteers to share their ideas with the class as you record them at the overhead. You may find it helpful to project the overhead image directly onto the whiteboard so that students can mark or loop the image to show how they are seeing the portion of the mat that is shaded. Number Corner, Grade 5 37

37 November Calendar Grid Fractions & Decimals (cont.) Students We said it s squares. We just colored in squares. But if the whole mat is worth, you d have to say that we colored in tenhundredths. But it s still of the little squares, right? We agree you could say ten-hundredths, but we also said that you can call it one-tenth of the mat because there are rows of and we just colored one of them in. Teacher Jaime, please come up to the whiteboard and show us where you re seeing ten-hundredths, and then let s have Morgan show how she and Kendra are thinking about one-tenth. Jaime You can just see that the square is split up into 0 little squares, so each one of them must be one-hundredth. We colored in of them, so we think it s ten-hundredths. Morgan But that row is also the same as of the whole square. See, if I loop it on the whiteboard, you can see that there are rows of going up and down. We only colored in of the rows, so that s a tenth of the square. Jon But I don t get it. There are 0 squares on the mat and we colored in of them, so why can t we just say? Yolanda Because the whole mat is only worth now instead of 0. Teacher Jaime says he colored in ten-hundredths of the square and Morgan says she colored in one-tenth of the square. Who do you agree with? Talk with the person next to you for a minute about this. 38 Number Corner, Grade 5

38 Calendar Grid Fractions & Decimals (cont.) November Students We think maybe the mat is kind of the same as a dime. You can say that a dime is pennies, and that s like ten-hundredths of a dollar. But you can also say that a dime is one-tenth of a dollar, right? If you look at the mat on the whiteboard, you can see that the part that s colored in is ten-hundredths and one-tenth. They re both right. It s like two different names for the same thing. Along with 0 and, a few students may reason that one column on the mat is the same as 2 nickels and might be written as If no one suggests a decimal name for the display, write a and a on the whiteboard. Restate that there are tiny squares or whole column of squares colored in, but the whole mat is worth. Then ask if students can think of a way to communicate the fact that the tiny squares or the whole column represent quantities less than without using words or fractions. Some students may recall that they can write one-tenth as 0.. If this does not come from the students, you might remind them that they can write one-tenth as 0., and then invite them to make sense of the decimal notation for one another. If students have drawn a comparison to a dime, you might instead ask them if they can think of how the amount of a dime would be shown on a price tag in the store. Finally, ask students to describe the last marker on the overhead in terms of money, fractions, and decimals. Equivalent Expressions Marker Money Fractions Decimals 2 3 dime cents ( ) It s worth pennies or 2 nickels. row on the mat is kind of like dime. They re both of the whole unit. Each square is like a dime because there are of them in the whole rectangle of a dollar of a dollar of a rectangle. $0.. (ten hundreths). (one tenth). (one tenth) At the end of the workout, let students know that before the next Calendar Grid Workout, you ll transfer the information from the overhead onto a large recording chart. After the second workout, calendar helpers will be responsible for updating both the Calendar Grid and the recording chart, but for now, they ll just need to insert a new marker on the grid each day. Note You ll need to prepare a large recording chart if you haven t already (see Advance Preparation on page 36). Before the second Calendar Grid Workout, attach Number Corner, Grade 5 39

39 November Calendar Grid Fractions & Decimals (cont.) the first three mini-markers and transfer the information from the overhead to the chart. If you plan to re-use the chart in future years, laminate it and use tacks or tape loops to attach the mini-markers and an overhead pen to record the equivalent expressions. The chart will serve as a visual focus throughout the month, and it may also provide a useful display later in the year when the class studies fractions and decimals. Week 2 Discussing Markers 4 7 Open your second workout by drawing students attention to the fourth marker, which shows two dimes. If necessary, move the marker off the calendar grid to a temporary location where everyone can see it clearly. Attach the corresponding mini-marker to the large recording chart and ask students to generate a variety of ways to name the coins in terms of money, decimals, and fractions. Record their responses directly on the chart. M arker & Date Equivalent Expressions Money Fractions Decimals dime cents ( ) It s worth pennies or 2 nickels of a dollar of a dollar. $0. 2 row on the mat is kind of like dime. They re both of the whole unit (ten hundreths). (one tenth) 3 Each square is like a dime because there are of them in the whole rectangle. of a rectangle. (one tenth) 4 2 dimes 20 4 nickels or Repeat this process with the fifth calendar marker, but this time, have students first replicate the marker by coloring in the second base ten mat on page 39 in their Student Books to match what you color in on the Calendar Grid Units overhead. Ask them to record two or three different expressions below the mat to show what part is colored in and then compare ideas with the people sitting nearest them. After they have had a minute to talk, ask volunteers to share their ideas. Project the overhead directly onto the whiteboard so students can mark or loop the mat as they explain how they see the 40 Number Corner, Grade 5

40 Calendar Grid Fractions & Decimals (cont.) November colored portion. As students share their ideas, record them on the chart, and encourage students to discuss any points of confusion that arise. November Overhead NC 3.4 Calendar Grid Units Money Base Ten Mat Kamela See the part I looped? That s one-tenth, so 2 of them must be two-tenths. Gabe We said the same thing, but we wrote it like a decimal. We wrote point-two because that means the same thing as two-tenths. Danny I m confused. Before, we said that each little square is one-hundredth, and we colored in 20 of them, so we wrote Is that the same as 2? Teacher Good question. What do the rest of you think? Are 2 and 20 0 the same? Josie We said yes because there are little squares that s hundredths in each strip, so 2 = 20. But there s another thing we re confused about. We tried to write the decimal for twenty-hundredths, and we got 0.020, but Yolanda wrote 0.2. Are those the same? Teacher When we write 0.0, what does that mean? Maria That s hundredth, like little square. Teacher So if we write 0.02, what does that mean? Kamil That s 2 little squares two-hundredths. I think if you want to write twenty-hundredths, you have to write it like this:.20 Gabe That s the same as what we wrote but it has a 0 after it. Teacher Talk to the person sitting next to you. Do these two numbers mean the same thing, and where can you see both of them on the mat?.2 and.20 Conduct a similar discussion about markers 6 and 7, again attaching the corresponding mini-markers to the chart and recording students ideas in the Number Corner, Grade 5 4

41 November Calendar Grid Fractions & Decimals (cont.) appropriate boxes. They may find it helpful to label the blank rectangle and pentagon on the Calendar Grid Units overhead to show how they see the parts that have been shaded in. M arker & Date Equivalent Expressions Money Fractions Decimals dime cents ( ) It s worth pennies or 2 nickels of a dollar of a dollar. $0. 2 row on the mat is kind of like dime. They re both of the whole unit (ten hundreths). (one tenth) 3 Each square is like a dime because there are of them in the whole rectangle. of a rectangle. (one tenth) 4 2 dimes 20 4 nickels or If the square was a dollar, the 2 rows could be 2 dimes or It s and of a 5 5 dollar is or if you split 5 each piece in half It s like 2 dimes Pentagon 2 Rectangle 5 5 = 2 42 Number Corner, Grade 5

42 Calendar Grid Fractions & Decimals (cont.) November Week 3 Completing and Discussing Number Corner Student Book page 53 Have student helpers update the Calendar Grid and the chart each day through the rest of the month. You may need to remind Monday s helpers that they re responsible for Saturday s and Sunday s markers as well. In the third week of the month, plan to conduct Calendar Grid Workouts on two consecutive days. On the first day, ask students to complete all but the last question on page 53 in their Number Corner Student Books. Although students will need to complete their own pages, they may find it helpful to work in pairs or table groups. Encourage students who are struggling to look at the recording chart for ideas. Assure them that they can also add to their sheets during the class discussion tomorrow. This material is an introduction to fractions and decimals, so don t be concerned if students understandings are fragile right now. Number Corner Student Book NAME DATE Thinking About Fractions & Decimals CALENDAR GRID Here are 4 of the markers from the calendar pattern this month. Label each one with at least 2 different fraction or decimal names List at least 2 ways in which the markers above are alike. 3 Darius says that all of these markers show 2 5 him? Why or why not? of something. Do you agree with 4 Here is the calendar marker for the 8th. Record both a fraction and a decimal to show what part of a dollar it is. Shade in the equivalent amount on the mat, the pentagon, and the rectangle. If it can t be done, explain why. Fraction: Decimal: 8 Have students complete the last question on this sheet after tomorrow s discussion. The following day, ask students to share the ways in which they described each marker. If you project the Calendar Grid Units overhead directly onto the whiteboard and have students use overhead coins and overhead pens to Number Corner, Grade 5 43

43 November Calendar Grid Fractions & Decimals (cont.) replicate markers 3, it will be easier for them to describe and loop the features of each marker that led to the fraction and decimal names they generated. Add new ways of describing each marker to the recording chart as they emerge. Mistakes or blank columns on the chart may also provide opportunities for discussion, controversy, and learning. At the end of the discussion, ask students to complete the last question sometime before next week s Calendar Grid Workout. (Students might do this as homework or in class during morning seatwork or at another time of day.) Week 4 Discussing Patterns Ask students to turn to page 53 in their Student Books, which they completed the previous week, and then draw their attention to the updated Calendar Grid. Ask the class to discuss and predict, first in pairs, and then as a group, what the next few markers will look like. As they share their predictions, ask them to explain their thinking. November Sunday Monday Tuesday Wednesday Thursday Friday Saturday Students I think the marker after today s will have a mat with 75 little squares colored in because 75 is 75 0 of a dollar. And then there will be a pentagon and then a rectangle. Are you sure? Sometimes the pentagon gets left out. It does? Doesn t it always go money, mat, pentagon, and then rectangle? No! Look at markers 8 and 9. Finally, have them pair-share their responses to the final question on Student Book page 53 and then call on volunteers to share their ideas with the class. If students have not noticed that some models are left out from time to time, they will now. Ask students to consider why 4 was not shown on either the 44 Number Corner, Grade 5

44 Calendar Grid Fractions & Decimals (cont.) November pentagon or the rectangle and whether they can think of any way to show 4 on either model. (Some students may have solved the problem by coloring in 2 sections on the pentagon and 2 2 squares on the rectangle.) You might also ask why the pentagon was not used to model or 2. Based on the patterns they can see so far, can they identify a quantity still to emerge on the calendar that won t be represented with all four models? Week 4 or 5 Discussing Markers or Number Corner Student Book page 58 If you have time to conduct one more workout at the end of the month, ask students to discuss markers Follow the steps described for week 2, and add to or edit the entries for markers on the recording chart if necessary. If most students are seeing the fractions and decimals easily, you might have them complete Student Book page 58 instead. Number Corner Student Book NAME DATE More About Fractions & Decimals CALENDAR GRID a Circle the fraction that is greater. 4 b Use labeled sketches, words, and/or numbers below to show that you are correct. You can use the base ten mats to help if you like. 2a Circle the fraction that is greater b Use labeled sketches, words, and/or numbers below to show that you are correct. You can use the base ten mats to help if you like. 3 Use labeled sketches, words, and/or numbers to add the two fractions below. You can use the base ten mat or the 2-by-5 array to help if you like = Number Corner, Grade 5 45

45 November Overhead NC 3.4 Calendar Grid Units Money Base Ten Mat Pentagon Rectangle Number Corner The Math Learning Center

46 November Blackline NC 3. Run copy and post beside the Calendar Grid. Calendar Grid Units The mat, pentagon, dollar, and array are each worth whole unit this month. unit unit unit unit The Math Learning Center Number Corner

47 November Blackline NC 3.2 Run copy, trim and attach to subsequent panels to create one long chart. Recording Chart Panels page of 2 Marker & Date Equivalent Expressions Money Fractions Decimals Number Corner The Math Learning Center

48 November Blackline NC 3.3 Run copies, trim and attach to subsequent panels to create one long chart. Recording Chart Panels page 2 of 2 The Math Learning Center Number Corner

49 November Blackline NC 3.4 Run copy and cut along the lines to create mini-markers. November Mini-Markers page of Number Corner The Math Learning Center

50 November Blackline NC 3.5 Run copy and cut along the lines to create mini-markers. November Mini-Markers page 2 of The Math Learning Center Number Corner

51 November Blackline NC 3.6 Run copy and cut along the lines to create mini-markers. November Mini-Markers page 3 of Number Corner The Math Learning Center

52 Number Corner Student Book NAME DATE Thinking About Fractions & Decimals CALENDAR GRID Here are four of the markers from the calendar pattern this month. Label each one with at least 2 different fraction or decimal names List at least 2 ways in which the markers above are alike. 3 Darius says that all of these markers show 2 5 him? Why or why not? of something. Do you agree with 4 Here is the calendar marker for the 8th. Record both a fraction and a decimal to show what part of a dollar it is. Shade in the equivalent amount on the mat, the pentagon, and the rectangle. If it can t be done, explain why. Fraction: Decimal: 8 The Math Learning Center Number Corner 53

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54 October Sunday Monday Tuesday Wednesday Thursday Friday Saturday Calendar Grid October Overhead NC 2.9 DATES What s in a Name? Line Plot CALENDAR COLLECTOR Number of Students (each x stands for student) Number of Letters in the First and Last Name Mode (number that appears most often in the set) Median (middle number when the numbers are put in order) Range (difference between the highest and lowest number in the set) Mean (average) = = = 2 Week 2 Weeks 3 Weeks Calendar Collector Number Corner, Grade 5 73

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56 October October Calendar Grid CALENDAR GRID Similar Figures Overview Each pair of calendar markers features two similar shapes: first a small version of the shape, and then a larger one. In each case, the lengths of the shape s sides are doubled in the larger version, resulting in a shape with an area 4 times greater than the area of the first. Frequency Update the Calendar Grid each day and share observations and predictions once or twice a week. Skills & Concepts H identifying and describing patterns H exploring the concept of similarity H investigating how a change in one variable relates to a change in a second H identifying, describing, and comparing polygons H developing methods for determining the areas of polygons You ll need H Mini Similar Shapes Markers (Overheads NC 2.2 and 2.3) H October Calendar Grid Record Sheet, pages and 2 (Blacklines NC 2.3 and 2.4, see Advance Preparation) H Geoboard Recording Paper (Blackline NC 2.5, class set, optional) H Taking a Closer Look at the Pattern (Number Corner Student Book, page 27) H Similar Figures (Number Corner Student Book, pages 33 and 34) H Calendar Grid pocket chart H Day, Month, and Year markers H Similar Shapes calendar markers H half-class set of geoboards and geobands H scissors H paper to mask portions of the transparency H math dictionaries (optional) Advance Preparation Run copy each of Blacklines NC 2.3 and 2.4. Trim and attach them to form one long record sheet. Post the record sheet on your display board before the first Calendar Grid Workout of the month. Background Information for the Teacher: Similar Figures Two figures are said to be similar if they have exactly the same shape. Similar figures may or may not be of the same size. (Two figures that are exactly the same shape and size are said to be congruent. Congruent figures are a subset of similar figures.) When identifying and constructing similar figures, it is helpful to keep in mind that all of the Number Corner, Grade 5 95

57 October Calendar Grid Similar Figures (cont.) side lengths must change by the same ratio from one figure to the next, while the angles must remain exactly the same. For example, if you double the lengths of all sides of the smaller rectangle below, the result is a similar rectangle. 4 cm cm 2 cm cm 2 cm 2 cm 2 cm Similar Rectangles 4 cm However, if you double one dimension while tripling the other, for example, the resulting rectangle is not similar to the first, as shown below. 6 cm cm 2 cm cm 2 cm 2 cm 2 cm 6 cm Not Similar Rectangles Week Introducing the New Markers and Record Sheet Invite a volunteer to post the calendar markers up to and including today s and give students a few minutes to examine the collection quietly. Then invite volunteers to describe what they see and make predictions about future markers. October Sunday Monday Tuesday Wednesday Thursday Friday Saturday Students There are 2 rectangles and then 2 triangles. There s a little one and a big one each time. 96 Number Corner, Grade 5

58 Calendar Grid Similar Figures (cont.) October The little one and the big one are both the same color. They all start in the corner. Maybe tomorrow there will be a different shape. I ll bet it ll be a little one, and the day after that will be the same one, only bigger. If students don t use the word similar, let them know that the shapes on markers and 2, and on markers 3 and 4, are similar, which means that they are exactly the same shape (although they are not the same size). Then draw students attention to the Calendar Grid Record Sheet, and, with input from the class, fill in the information up to the present date. Before having students identify the area of each figure, explain that the smallest square on the grid is square unit of area. If students cannot see the markers clearly from where they sit, you might want to display the first Mini Similar Shapes Markers overhead. Be sure to mask the other figures on the transparency so as not to ruin the surprise of future markers. Once the record sheet has been brought up to date, ask students to share further observations and predictions. October Overhead NC 2.2 Mini Similar Shapes Markers page of Date October Calendar Grid Record Sheet Shape Name Area in square units rectangle sq. unit 2 2 rectangle 2 sq. units 3 4 right triangle right triangle 2 sq. unit 2 sq. units Other Observations It s really little, and just takes up half a square unit. This shape looks the same as the first one. It s similar but bigger. It s easy to see that this shape is half a square unit. This triangle takes up one whole square and 2 halves. Finally, explain that it will be the student helpers responsibility to post a new calendar marker and update the record sheet each day. Make sure students know how they can find information if they are not sure what the name of a particular shape is (e.g., an illustrated math dictionary). For your own reference, the names of the shapes and more information about triangle classification are included on page. Week 2 Completing and Discussing Number Corner Student Book Page 27 In the second week of the month, plan to conduct Calendar Grid Workouts on two consecutive days. On the first day, have students complete page 27 in their Number Corner Student Books. Make Geoboard Recording Paper and geoboards available so that students can use them to find the areas of some figures if needed. Cover the Calendar Grid Record Sheet before they get started so that students can practice finding the area of different shapes themselves instead of copying the information. Invite students to brainstorm Number Corner, Grade 5 97

59 October Calendar Grid Similar Figures (cont.) observations for problem 2 in pairs if they like. Others may prefer to work alone. You might help students focus their work by encouraging them to look for likenesses and differences between each pair of shapes. You might also encourage students to think about how the side lengths and area of each pair of similar shapes are related. Answers, including sample observations, are provided in the October Answer Key beginning on page 5. Number Corner Student Book NAME DATE Taking A Closer Look at the Pattern CALENDAR GRID Fill in the chart below to show the name of each shape above and its area in square units. Date Shape Name Area in square units = square unit Key 2 List at least 4 different mathematical observations about the pattern so far. If some students have trouble with the triangles on markers 7 and 8, you may want to invite several students to show how they found the area of each triangle using the Mini Similar Shapes Markers overhead. Challenge students to use the Geoboard Recording Paper and the geoboards and geobands to prove their assertions about the figures. 7 Jade I imagined cutting the tip of the triangle part off, turning it around, and fitting it in on the bottom. You can see it would fill square. Armin I drew a box around the rectangle the triangle s in. Then I could see that the triangle fills half of it. Since the rectangle is 2 squares big, the triangle is half of that:. 98 Number Corner, Grade 5 Teacher Jade says that you can cut off the tip of the triangle and fit it with the bottom of the triangle to make one square. Armin said that the

60 Calendar Grid Similar Figures (cont.) October triangle makes up half of this two-square rectangle, so it has to have an area of. Please spend some time talking to the person next to you about how you can tell whether or not these two ideas are true or not. You can use the Geoboard Recording Paper or the geoboards to explore the ideas. Then we ll share as a group. Rian We thought about Armin s idea. If the two triangles are the same, then they are each half of the rectangle. So Yolanda and I made the rectangle and triangle on a geoboard, and so did Gabe and Danny. Then we put them on top of each other. We could see the triangles were the same. So each one is half of the rectangle. Yolanda Right, so since the rectangle is 2, each triangle is. Omid I thought about Jade s idea, because if I could see it all in square, that would make more sense to me. So I drew the triangle on the geoboard paper and then I cut the tip off. You can flip it around and it fits right in there, see? It s square when you fit it together like that. The following day, ask each student or pair of students to volunteer one observation they made about the pattern. Challenge them to make it all the way Number Corner, Grade 5 99

61 October Calendar Grid Similar Figures (cont.) around the room without repeating any observations. Afterward, students can share additional observations and predictions as time allows. Week 3 Discussing the Pattern and the Shapes During the third week, take at least one workout to have students discuss the pattern. Encourage students to talk about the names and properties of each shape, as well as the way in which each small shape has been enlarged to create a similar figure. What happens to the angles, the length of each side, and the area when the smaller of each pair is enlarged? What would happen if the grids were larger and each figure could be enlarged a third time? What patterns can they see in the number of square units as the month progresses? Week 4 Completing and Discussing Number Corner Student Book Pages 33 and 34 Conclude the month with two back-to-back Calendar Grid Workouts in which students work in pairs to complete pages 33 and 34 in their Student Books the first day and then discuss their work the next day. When they share their work, ask them to bring their geoboards to the overhead to show everyone their shapes and prove that they are similar. If disagreements arise, you may need to remind students to disagree respectfully and then challenge them to prove their thinking using the geoboards. Number Corner Student Book Number Corner Student Book NAME DATE NAME DATE Similar Figures page of 2 Similar Figures page 2 of 2 CALENDAR GRID CALENDAR GRID When people say two things are similar, they mean that they re alike in many ways. The word similar has a very specific meaning in geometry. Similar figures have exactly the same shape, but are usually different sizes. Fill in the oval under the pair or pairs of shapes that are similar. 3 If the smallest square on the geoboard is unit of area, what is the area of each shape you drew on page 33. = square unit a Pair, Shape A Pair, Shape B b Pair 2, Shape A Pair 2, Shape B c Pair 3, Shape A Pair 2, Shape B 2 Work with a partner to make three pairs of similar figures on a geoboard. Record your work below. Example Pair CHALLENGE 4 If you make 2 similar figures on a geoboard, what do you have to do to the sides of the first figure to get the area of the second figure to be exactly 4 times more than the area of the first one? A B A B 0 Number Corner, Grade 5

62 October Calendar Grid Similar Figures (cont.) KEY TO SHAPE NAMES Marker Pair Shape Name and 2 rectangle 3 and 4 isosceles right triangle 5 and 6 parallelogram 7 and 8 scalene right triangle 9 and trapezoid and 2 pentagon 3 and 4 square 5 and 6 isosceles triangle 7 and 8 quadrilateral 9 and 20 pentagon 2 and 22 trapezoid 23 and 24 pentagon 25 and 26 hexagon 27 and 28 pentagon 29 and 30 square 3 scalene right triangle Note You and the class may notice that the figures alternate between having an even number of sides and an odd number of sides. Background for the Teacher The triangles on this month s calendar grid fall into a few categories. The definitions below will help you make sense of how each has been categorized in the table above. One way to classify triangles is by the relationships among their sides. In an equilateral triangle, all three sides are of equal length. In an isosceles triangle, just two sides are of equal length. A scalene triangle has three sides of different lengths. The side lengths of triangles are also related to the angles within them. In an equilateral triangle, all three angles are equal (60º). In an isosceles triangle, two angles are equal. In a scalene triangle, no angles are equal. As you can see, these categories do not overlap. B E H A C D F G I equilateral All sides are equal. All angles are equal. isosceles Sides DE and EF are equal. Angles D and F are equal. scalene No sides are equal. No angles are equal. A right triangle contains one right angle (90º). Therefore, it is not possible for a right triangle to be equilateral. However, right triangles may be isosceles or scalene, Number Corner, Grade 5

63 October Calendar Grid Similar Figures (cont.) as shown below. The triangle on the left is similar to the triangles on markers 3 and 4, and the triangle on the right is similar to the triangles on markers 7 and 8. K Q J isosceles right triangle J is a right angle. Angles K and L are both 45, and side JK is equal to side JL. L P scalene right triangle P is a right angle. None of the sides are equal. None of the angles are equal. R 2 Number Corner, Grade 5

64 October Overhead NC 2.2 Mini Similar Shapes Markers page of Number Corner The Math Learning Center

65 October Overhead NC 2.3 Mini Similar Shapes Markers page 2 of The Math Learning Center Number Corner

66 October Blackline NC 2.3 Run copy, trim, attach to Blackline NC 2.4, and post on your calendar display. October Calendar Grid Record Sheet page of 2 October Calendar Grid Record Sheet Date Shape Name Area in square units Other Observations Tape or glue page 2 here. The Math Learning Center Number Corner

67 October Blackline NC 2.4 Run copy, trim, attach to Blackline NC 2.3, and post on your calendar display. October Calendar Grid Record Sheet page 2 of 2 Number Corner The Math Learning Center

68 October Blackline NC 2.5 Run as needed. Geoboard Recording Paper The Math Learning Center Number Corner

69 Number Corner Student Book NAME DATE Taking a Closer Look at the Pattern CALENDAR GRID Fill in the chart below to show the name of each shape above and its area in square units. Date Shape Name Area in Square Units = square unit Key 2 List at least four different mathematical observations about the pattern so far. The Math Learning Center Number Corner 27

70 Number Corner Student Book NAME DATE Similar Figures page of 2 CALENDAR GRID When people say two things are similar, they mean that they re alike in many ways. The word similar has a very specific meaning in geometry. Similar figures have exactly the same shape, but are usually different sizes. Fill in the oval under the pair or pairs of shapes that are similar. 2 Work with a partner to make three pairs of similar figures on a geoboard. Record your work below. Example Pair A B A B Pair 2 Pair 3 A B A B (Continued on back.) The Math Learning Center Number Corner 33

71 Number Corner Student Book NAME DATE Similar Figures page 2 of 2 CALENDAR GRID 3 If the smallest square on the geoboard is unit of area, what is the area of each shape you drew on page 33. = square unit a Pair, Shape A Pair, Shape B b Pair 2, Shape A Pair 2, Shape B c Pair 3, Shape A Pair 3, Shape B CHALLENGE 4 If you make two similar figures on a geoboard, what do you have to do to the sides of the first figure to get the area of the second figure to be exactly 4 times more than the area of the first one? 34 Number Corner The Math Learning Center

72 Two-Penny Toss Record Sheet Day 2 Heads (2H) Head/ Tails (H/T) 2 Tails (2T) Running Totals 2H H/T 2T Total Tosses Calendar Collector September Sunday Monday Tuesday Wednesday Thursday Friday Saturday Calendar Grid Number Corner, Grade 5 23

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74 August & September August & September Calendar Collector CALENDAR COLLECTOR The Two-Penny Toss Overview The first Calendar Collector is an extended probability experiment in which students flip two pennies times for each day in September or 30 out of 3 days if you re conducting the workout in August. Before they begin, students determine what the possible results are and make predictions about the likely outcomes of 300 flips. They record all of the results throughout the month and, based on that data, revise their predictions about the likelihood of each possible outcome at mid-month and again at the end of the month. Frequency Update the data daily, and share observations and predictions about the data as a whole group once a week. Skills & Concepts H predicting and representing all possible outcomes for a simple probability situation H describing the probability of various outcomes or events H collecting, recording, and interpreting data H exploring basic concepts of sampling You ll need H Two-Penny Toss Data Chart (Overhead NC.6) H Two-Penny Toss Record Sheet, pages and 2 (Blacklines NC.2 and.3, copy each, trimmed, attached, and posted on the Number Corner display, see Advance Preparation) H Two-Penny Toss Data Chart (Number Corner Student Book, page 4) H Thinking about the Two-Penny Toss Experiment (Number Corner Student Book, page 9) H One More Look at the Two-Penny Toss Data (Number Corner Student Book, page 9) H 2 pennies H overhead marking pens H calculators Advance Preparation You may prefer to run copies of this record sheet on transparencies. You can post them on a white paper background with reusable adhesive putty. That way, students can record directly on the transparency, which you can easily remove to share at the overhead. H constructing, reading, and interpreting bar graphs Week Introducing the Two-Penny Toss Experiment Introduce this month s Calendar Collector by showing students two pennies. Ask them what the possible outcomes are if you toss or flip both at the same time. Give them a moment to think quietly, and then list the possibilities on the board as students share them. Students will probably identify three possi- 40 Number Corner, Grade 5

75 Calendar Collector The Two-Penny Toss (cont.) August & September bilities: both will come up heads, both will come up tails, and one will come up heads while the other will come up tails. Next, ask students to consider the relative likelihoods of these outcomes. You might ask them to imagine that they flipped both pennies 0 times. Would one or more of the possibilities have a greater chance of occurring, or are the three equally likely? Ask them to explain their thinking. Most students will probably reason that because there appear to be three possibilities, each has the same out of 3 chance of occurring. However, a few students might realize that because there are 2 different pennies, the chances of getting heads/ tails far outweigh the chances of getting two of a kind. If students debate this issue, encourage them to explain and justify their assertions without suggesting which is correct. Don t be concerned if this debate does not arise at this time: students will begin to rethink their initial predictions as they review the data throughout the month. After students have had a chance to discuss the possible outcomes, explain that they will conduct trials for each day in September (or 30 out of the 3 days if you re doing the workout in August). Ask them to think about the following questions: How many trials will they conduct by the end of the month? ( 30 days = 300) Do they really need to collect that much data? How much do they think is enough to determine the chances of getting 2 heads, 2 tails, or one of each? Based on how many days have passed already this month, how many tosses will they need to make today? Finally, have volunteers flip the 2 pennies and report the results as you record the data on the Calendar Collector Record Sheet. You might call students up one by one or you might have them pass the 2 pennies around the room. Each time the coins are tossed and the results announced, make a tally mark in the appropriate box on the record sheet. After tosses have been made, record the totals at the end of the first row. Repeat this procedure as many times as needed to bring the record sheet up to date. When you fill in the values at the end of the second row and beyond, be sure students understand that these are running, or cumulative, totals that reflect the total number of times each outcome has been obtained so far this month. Day 2 Heads (2H) Two-Penny Toss Record Sheet Head/ Tails (H/T) 2 Tails (2T) Running Totals 2H H/T 2T Total Tosses Number Corner, Grade 5 4

76 August & September Calendar Collector The Two-Penny Toss (cont.) Jasmine Why are you writing 3, 9, and 8 at the end of the line when we got both heads twice, heads/ tails 5 times, and 2 tails 3 times for the second day? Teacher These are our running totals. Each day, the numbers at the end of the row need to show how many times we got each outcome for the whole month altogether. In all, we ve gotten both heads 3 times, heads/ tails 9 times, and both tails 8 times. How do these totals compare with your predictions? Jon I think those pennies are trick pennies that come up with more tails than heads! Theo We re getting less both heads and more of the other two than we thought we would. Conclude the workout by collecting any more data needed to bring the record sheet up to date, modeling the recording process again. After the necessary number of tosses have been made, recorded, and totaled, ask students to share any observations and predictions they can make about the likely outcomes of this experiment over the course of the month. Based on the data above, some fifth-graders would say that you re more likely to get head/ tail than 2 heads or 2 tails, but may be quite puzzled as to why. Others may be convinced that all three outcomes are equally likely and that the results simply reflect the fact that they haven t collected much data yet. If so, ask them how many more days they think will need to pass before they can be sure about the accuracy of their predictions. After conducting the first workout, be sure that the record sheet is updated each day and that on Mondays your student helpers understand that they need to make and record tosses for the weekend days as well. (If days get skipped during the week, student helpers can collect and record the data for the missing days as they have time.) 42 Number Corner, Grade 5

77 Calendar Collector The Two-Penny Toss (cont.) August & September Week 2 Compiling & Considering the Data Collected So Far Before you conduct the second workout, make sure the record sheet is up to date and enter the running totals on the Two-Penny Toss Data Chart overhead. Day 2 Heads (2H) Two-Penny Toss Record Sheet Head/ Tails (H/T) 2 Tails (2T) Running Totals 2H H/T 2T Total Tosses August & September Overhead NC.6 (Accompanies Number Corner Student Book pages 4 and 9.) Two-Penny Toss Data Chart CALENDAR COLLECTOR Date 2 Heads (2H) Head/ Tails (H/T) 2 Tails (2T) Total Number of Tosses 9/ Decide on a scale for this bar graph that will be big enough to fit the data for the entire month. Then label both axes. 2 Show the data from the chart above on the bar graph. Label it carefully so other people can understand it. 3 Based on the data the class has collected so far, how do you think this experiment is going to turn out by the end of Display the Two-Penny Toss Data Chart and ask students to think for a minute about how the results so far compare with their original predictions and then share their ideas with a partner. Take a couple of minutes for students to share any observations and conjectures they have with the class, and then review and discuss the three tasks listed on the transparency. Depending on your students previous experiences with graphs, you may need to spend a few minutes talking about how to establish an appropriate scale and label the graph. There are 23 cells in each column, and the scale will have to accommodate the data for the entire month, so each cell will have to stand for more than trial. There are a number of appropriate scales for this bar graph, and two examples are shown on the next page. As you work with your students, keep in mind that no column will have to go all the way to 300, but some will certainly need to go higher than 50. Both graphs on the next page are also filled in as examples. Don t fill in the bar graph on the overhead; students will do this on their own. Number Corner, Grade 5 43

78 August & September Calendar Collector The Two-Penny Toss (cont.) Number of Times We Got Each Outcome H H/T Outcomes 2T How Many Times the Pennies Landed That Way H H/T 2T How the Pennies Landed appropriate scale appropriate scale 2 Once they understand the tasks, have students complete Number Corner Student Book page 4, which is identical to the Two-Penny Toss Data Chart overhead. Note If student interest begins to wane partway into the month, you can use an alternative way to collect data. Instead of having helpers do it daily, break the class into small groups and ask each group to be responsible for collecting two days data. Then ask students to share their results, which can be compiled on the record sheet. Week 3 Analyzing the Data Before you conduct the third Calendar Collector Workout, transfer the data from the Two-Penny Toss Record Sheet to the Two-Penny Toss Data Chart overhead, just as you did before the second workout. August & September Overhead NC.6 Two-Penny Toss Data Chart CALENDAR COLLECTOR Date 2 Heads (2H) Head/ Tails (H/T) 2 Tails (2T) Total Number of Tosses 9/ Display this information at the overhead to begin the third workout. By now, it will be apparent that the two pennies are coming up heads/ tails far more frequently than they are both heads or both tails, but students may feel puzzled about why this is so. Take a couple of minutes for students to share 44 Number Corner, Grade 5

79 Calendar Collector The Two-Penny Toss (cont.) August & September any observations and conjectures they have, and then ask them to complete page 9 in their Number Corner Student Books. Number Corner Student Book NAME DATE Thinking about the Two-Penny Toss Experiment CALENDAR COLLECTOR Date 2 Heads (2H) Head/ Tails (H/T) 2 Tails (2T) Total Number of Tosses About what fraction of the total number of tosses is represented by each outcome? (For instance, if you made 60 tosses so far and you got 2 heads 4 times, that s about 4 of the tosses, because 40 is 4 of 60 and 4 is very close to 40.) a The pennies have both come up heads about of the time so far. b The pennies have come up heads/ tails about of the time so far. c The pennies have both come up tails about of the time so far. 2 To calculate probability, you need to make a list of all the possible outcomes. This list is called a sample space. One way to make a sample space is to use a chart like the one in the next column. Use this chart to show all the possible outcomes of the two-penny toss experiment. (The first one is done for you.) 2 (continued) Second Penny Heads Tails First Penny Heads Tails HH 3 According to the chart above, what is the probability of getting each outcome? a 2 heads b head and tail c 2 tails 4 Use the information on the chart to help explain why the two-penny toss experiment is turning out the way it is. After completing the chart in question 2, some students may observe that there are actually four possible outcomes, not three. Each of the four possibilities has an equal chance of occurring, and because there are 2 different ways to get a head/ tail outcome, that outcome occurs about twice as many times as each of the other two outcomes. While most fifth graders will not explain the situation this succinctly, this page will provide students with some insights about why the experiment is turning out the way it is. Week 4 Compiling & Drawing Conclusions about the Data Wait until as late in the month as possible to conduct the fourth workout so that students can consider the outcomes of a larger number of trials. Before conducting the last Calendar Collector Workout this month, transfer the data from the Two-Penny Toss Record Sheet to the Two-Penny Toss Data Chart overhead. Display the overhead and invite students to share observations about the data. Then ask students to think about the totals in relation to one another. About what fraction of the time did each result occur? Students will probably find that after making about 280 or so tosses, about one-quarter of the time they got 2 heads, about one-quarter of the time they got 2 tails, and about half the Number Corner, Grade 5 45

80 August & September Calendar Collector The Two-Penny Toss (cont.) time they got heads and tails. Invite students to use calculators to help if needed. Then have students enter the data on page 9 of their Student Books and answer the questions on the page using the data. If there are still a few days remaining in the month, ask students to think about whether the additional data is likely to alter their conclusions. Number Corner Student Book NAME DATE One More Look at the Two-Penny Toss Data CALENDAR COLLECTOR Date 2 Heads (2H) Head/ Tails (H/T) 2 Tails (2T) Total Number of Tosses Circle the graph that comes closest to showing the results of your experiment so far. a b H/T H/T 2H 2T 2H 2T c d 2H 2T H/T 2H H/T 2T 2 Explain why you chose the graph you did. Have student helpers continue to collect and record data until the last day of the month (or the 30th if you are conducting this workout in August). If there is time and student interest is high, take a few minutes at the very end of the month to re-examine the totals one more time. 46 Number Corner, Grade 5

81 August & September Blackline NC.2 Run copy, trim, attach to page 2, and post on the calendar display. Two-Penny Toss Record Sheet page of 2 Two-Penny Toss Record Sheet Day 2 Heads (2H) Head/ Tails (H/T) 2 Tails (2T) Running Totals 2H H/T 2T Total Tosses Glue or tape page 2 here. Number Corner The Math Learning Center

82 August & September Blackline NC.3 Run copy, trim, attach to page, and post on the calendar display. Two-Penny Toss Record Sheet page 2 of 2 The Math Learning Center Number Corner

83 Number Corner Student Book NAME DATE Two-Penny Toss Data Chart CALENDAR COLLECTOR Date 2 Heads (2H) Head/ Tails (H/T) 2 Tails (2T) Total Number of Tosses Decide on a scale for this bar graph that will be big enough to fit the data for the entire month. Then label both axes. 2 Show the data from the chart above on the bar graph. Label it carefully so other people can understand it. 3 Based on the data the class has collected so far, how do you think this experiment is going to turn out by the end of the month? Why do you think it s working out this way? 4 Number Corner The Math Learning Center

84 Number Corner Student Book NAME DATE Thinking about the Two-Penny Toss Experiment CALENDAR COLLECTOR Date 2 Heads (2H) Head/ Tails (H/T) 2 Tails (2T) Total Number of Tosses About what fraction of the total number of tosses is represented by each outcome? (For instance, if you made 60 tosses so far and you got 2 heads 4 times, that s about 4 of the tosses, because 40 is 4 of 60 and 4 is very close to 40.) a The pennies have both come up heads about of the time so far. b The pennies have come up heads/ tails about of the time so far. c The pennies have both come up tails about of the time so far. 2 To calculate probability, you need to make a list of all the possible outcomes. This list is called a sample space. One way to make a sample space is to use a chart like the one in the next column. Use this chart to show all the possible outcomes of the two-penny toss experiment. (The first one is done for you.) 2 (continued) Second Penny Heads Tails First Penny Heads HH Tails 3 According to the chart above, what is the probability of getting each outcome? a 2 heads b head and tail c 2 tails 4 Use the information on the chart to help explain why the two-penny toss experiment is turning out the way it is. The Math Learning Center Number Corner 9

85 Number Corner Student Book NAME DATE One More Look at the Two-Penny Toss Data CALENDAR COLLECTOR Date 2 Heads (2H) Head/ Tails (H/T) 2 Tails (2T) Total Number of Tosses Circle the graph that comes closest to showing the results of your experiment so far. a 2H H/T 2T b 2H H/T 2T c d 2H 2T H/T 2H H/T 2T 2 Explain why you chose the graph you did. The Math Learning Center Number Corner 9

86 March Overhead NC 7.5 March Meadow Grid CALENDAR GRID 2 Sunday Monday Tuesday Wednesday Thursday Friday Saturday Feeding Area B From Nest, Go West 2, Northwest 4 Nest 2 From Nest, Go South 4 5 From Nest 2, Go North, Northeast CLUE 2: A, A Feeding Area 2A 6 Feeding Area 2B From Nest 2, Go North, Northwest 7 Starting Point (0,7) Nest 3 From Nest 2, Go East 4 8 Nest From Starting Point, Go East 4 From Nest 3, Go West 2, Southeast CLUE 3: C, C Feeding Area 3A 2 9 Feeding Area A From Nest, Go West 2, Northeast CLUE : 2 WORDS B 2B A Net 2 Net 2A 3B 3A Net (4,7) Nest Nest 2 Nest 3 Nest 4 Nest 5 F.A. A F.A. 2A F.A. 3A F.A. 4A F.A. 5A F.A. B F.A. 2B F.A. 3B F.A. 4B F.A. 5B (3,8) (,8) (4,3) (5,5) (3,5) (8,3) (7,2) (5,2) Nest 6 Nest 7 Nest 8 Nest 9 Nest F.A. 6A F.A. 7A F.A. 8A F.A. 9A F.A. A F.A. 6B F.A. 7B F.A. 8B F.A. 9B F.A. B Calendar Grid Treasure Clues 2 words A, A, C, C March Overhead NC 7.2 Mrs. L Blue Put It on the Line, 2 Game What the 4 is in the value 75.47? of If, the the large square what is the value shaded part? is of Tanner was in walk- a-thon at his school. When he saw his mom after school he, said, walked 2.3 miles. If I d w alked just _ mile more, would have walked 3.0 miles in all. Fill in the missing number. I S handra ate 4 sandwich for a nd for lunch. much does she have left? of snack her How still 8.5 = 0.5 Computational Fluency What is the decimal n umber that repre - sents the shaded part of this pentagon? 0 0 Imani got chocolate ate 0 of it How much have left? king-si ed bar. She right away. does she a. 0.3 c I a 4 a Number Corner, Grade 5 285

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88 March March Computational Fluency COMPUTATIONAL FLUENCY Put It on the Line with Fractions & Decimals Overview This month s Computational Fluency Workouts return to the game of Put It on the Line. In this version, students work with decimals, fractions, and mixed numbers between 0 and 2.5. Frequency Once a week Skills & Concepts H using place value to read, write, compare, and order decimals through the hundredths place H reading, writing, comparing, and ordering fractions H locating decimals and fractions on a number line H generating equivalent fractions (30/0 = 3/; 3/5 = 6/) H using models to relate decimals to fractions that name tenths and hundredths H using concrete objects, pictures, words, and numbers to add and subtract fractions with like denominators H adding and subtracting decimals You ll need H Put It on the Line, Games 4 (Overheads NC 7.2, 7.9, 7.2, and 7.4, see Advance Preparation) H Put It on the Line, Games 4 (Number Corner Student Book, pages 29 and 38) H calculators H half-class set of base ten pieces (Run Blackline NC.4 on cardstock and cut apart to make your own base ten pieces as needed.) H 2 2 sticky notes (see Advance Preparation) H overhead pens in red and blue H colored pencils Advance Preparation Before each Computational Fluency Workout, cover each of the problems at the bottom of the appropriate transparency with a stack of 3 or 4 sticky notes so that the problems are not visible as the light shines up through the transparency. You can re-use the sticky notes from week to week. Put It on the Line, Game 0. 0 March Overhead NC 7.2 What is the value of the 4 in 75.47? If, the the what large is square the value shaded part? is of Tanner a-thon When after was at he his school walked 2.3 w alked just more, walked 3.0 Fill in I the number. in walka school. saw his he, miles. mom said, If would have miles missing I I d mile in all. Shandra ate of her 4 sandwich for snack and 4 much have for does left? lunch. she How still 8.5 = 0.5 What is the decimal number that repre- sents the shaded part of this pentagon? 0 Imani got a king-si ed chocolate bar. She ate of it right away. How much does she have left? a. 0.3 c b. 0. d. 3.5 Number Corner, Grade 5 299

89 March Computational Fluency Put It on the Line with Fractions & Decimals (cont.) Note There are four variations of the game offered this month. If your Spring Break comes at the end of March, you may want to have students play the fourth game at the beginning of April. An alternative would be to select the three variations that best meet the needs of your students. Each variation is slightly more challenging than the one before it. Using Base Ten Pieces and Calculators to Make Sense of Fractions and Decimals While some students may be able to solve most or all of the problems mentally, make base ten pieces available at each table or cluster of desks so students can use them to model situations in the game if they like. Doing so will greatly enhance their understandings of fractions and decimals. The pieces are named differently and assigned different values to represent decimal numbers. If students have only used the pieces to represent whole numbers before, take some time to review the names and values of the pieces. unit striplet 0. matlet 0.0 While some students will easily understand or may already know that 8 is equal to 0.8 or 0.80, the base ten pieces make it possible to see how and why. Blanca I turned some of my striplets over so you don t have to see all the little lines and put them on top of my mat. You can see that they fill up 8 of the mat. Jade And also, since the mat has 0 little matlets on it, you can see that 8 striplets cover up 80 0 of the mat. Morgan So if you remember that the first number on the right side of the decimal point is tenths and the second one is hundredths, it s easy to see why you can say that 8 is the same as 0.8 or Number Corner, Grade 5

90 Computational Fluency Put It on the Line with Fractions & Decimals (cont.) March The base ten pieces may also be instrumental in helping students make some of the calculations with greater understanding. While some may already know that 0. is.00, the pieces make it possible to display the quantity 0. and to see that ten times this quantity is, in fact,. 0. = 0 0. =.0 0 = 0 0 Calculators may also be a source of instruction for students when it comes to decimal notation. While the base ten pieces help students see and understand the connection between fractions and decimals, the calculator serves to reinforce the link. 0. Jasmine Hey look! I punched in divided by 0 and I got 0.. Gerard That makes sense. is, so it makes sense that 0 would be one-tenth. Most fifth graders have seen and worked with decimals in relation to money, but not all of them yet understand that money amounts less than a dollar are actually fractions of a dollar, and perhaps even fewer understand that decimals are, in fact, fractions written in a different form. In this workout, students have the opportunity to see and understand that the dollars and cents notation used to write 30 cents ($0.30) means no dollars, 3 tenths of a dollar, and no hundredths. Furthermore, students will have repeated opportunities to see that 30 cents can also be represented using the fraction 3, meaning 3 tenths of a dollar, or 30 0, meaning 30 hundredths of a dollar, and that both fractions can also be written in decimal form as 0.3 or Number Corner, Grade 5 30

91 March Computational Fluency Put It on the Line with Fractions & Decimals (cont.) Week Playing Put It on the Line, Game Open the workout by displaying the Game transparency. Take a minute to have students discuss what they notice about the number line. Put It on the Line, Game 0. 0 March Overhead NC 7.2 Next, explain that you ll play class versus teacher. With each turn, you or they will uncover one problem at the bottom of the transparency. After the problem has been uncovered, you ll work together to solve it, and then the team that uncovered it will write the answer along the number line in their own color. At the end of the game, you ll add up each team s numbers, and the higher score will win. After explaining how the game is played, have students find the corresponding page in their Student Books. (Games and 2 are on page 29, while games 3 and 4 are on page 38.) Explain that they will mark the answers for each team in their books, using 2 different colored pencils, as the game proceeds. Finally, decide which team will be red and which will be blue. Note You ll find a key to all of the problems on this month s overheads on page 304. Best Practice Tip Asking students to paraphrase others ideas sends the message that they need to listen carefully to one another. It also helps them understand other strategies for solving a problem. When students solve the problems, give everyone the benefit of some think time. Ask them to read the problem together and then simply give the thumbsup sign when they have the answer, rather than raising their hands or calling out. Encourage students to use base ten pieces, calculators, sketches and/or paper and pencil computations to help make sense of the situation, with the understanding that they ll need to be able to explain their answers. When you see plenty of thumbs up around the room, have students pairshare their answers and reasoning. After a few moments, have students share their answers and thinking as a whole class. Whether students all agree on a particular answer or offer more than one solution, have students explain their thinking, using the pictures on the overhead as necessary. This is a chance for students to learn from one another, so encourage them to listen carefully. At times, you may even want to invite them to restate one another s strategies. number. number..5 = 0.5 What is the decimal number that repre- sents the shaded part of this pentagon? 0 Imani got a king-si ed chocolate bar. She ate of it right away. How much does she have left? = 0.5 What is the decimal number that repre- sents the shaded part of this pentagon? 0 Imani got a king-sized chocolate bar. She ate of it right away. How much does she have left? a. 0.3 c a. 0.3 c b. 0. d. 3.5 b. 0.6 d Number Corner, Grade 5

92 Computational Fluency Put It on the Line with Fractions & Decimals (cont.) March Kamela On this one we said the answer is six-tenths. The pentagon is cut into fifths. So if you split the fifths in half, you get tenths. Then just count them. There are six-tenths shaded in, because each fifth is like two-tenths. Take turns with the students to uncover a new problem. Use the procedure described above to have students solve each problem, and record the solution in the appropriate team s color on the overhead, while students do the same in their books. When all problems have been uncovered and solved, and the answers recorded, ask students to estimate each team s total. Mrs. L Blue Put It on the Line, Game March Overhead NC 7.2 What is the value of the 4 in 75.47? If the large square is, what is the value of the shaded part? Tanner was in a walk- Shandra ate of her 4 a-thon at his school. sandwich for snack When he saw his mom and 4 for lunch. How after school, he said I much does she still walked 2.3 miles. If I d have left? w alked just mile more, I would have walked 3.0 miles in all. Fill in the missing number. 8 Then ask students to find the exact totals for each team, again using their base ten pieces and calculators as needed. Once they have found and discussed the totals, take some time to review how decimals and fractions are added, using the base ten pieces to help. For example, there may be more than a few students in your class who believe that is 5 20, and the base ten pieces provide a way to demonstrate clearly why this is not the case. You may wish to have students record the computation in one or both formats on their own record sheets before concluding the workout NAME Put It on the Line! Game and 2 Put It Jaime COMPUTATIONAL 0. FLUENCY On the Line! Game = DATE 2/ = T eam Total _ 2 or 2.5T eam 2 Total 3.0 _ = = Number Corner Student Book Continuing Through the Month Each of the other three games is played the same way, although the numbers at the end of the line vary a bit. The first two games run from 0 to.0. The Number Corner, Grade 5 303

93 March Computational Fluency Put It on the Line with Fractions & Decimals (cont.) third game runs from 0 to.00, providing the opportunity to consider hundredths as well as tenths. The fourth game runs from 0 to 2.5. With 9 equal divisions marked between the first and last number, students will be working in increments of 0.25 on the last sheet. You ll probably discover in the process of playing these games that many of the problems are easy for some of your students, while others may not have made much sense of decimals and fractions yet. The scaffolding provided by the base ten pieces, the calculators, and the thinking of students who better understand these numbers will help all students build stronger decimal and fraction sense. Note Some teachers like to vary the game later in the month by dividing the class in half and having students play in two teams against one another rather than against the teacher. ANSWER KEYS FOR PUT IT ON THE LINE, GAMES 4 Answer Game Game 2 Game 3 Game or or or or or 3 0. or 0.70 or or or or or or or or or or or or or or 7 0. or or or or or or or 0.4 or or or or or or or or or or Number Corner, Grade 5

94 March Overhead NC 7.2 (Accompanies Number Corner Student Book page 29.) Put It on the Line, Game 0.0 What is the value of the 4 in 75.47? If the large square is, what is the value of the shaded part? Tanner was in a walka-thon at his school. When he saw his mom after school, he said I walked 2.3 miles. If I d walked just mile more, I would have walked 3.0 miles in all. Fill in the missing number. Shandra ate of her 4 sandwich for snack and 4 for lunch. How much does she still have left? 8.5 = 0.5 What is the decimal number that represents the shaded part of this pentagon? 0 Imani got a king-sized chocolate bar. She ate of it right away. How much does she have left? a. 0.3 c b. 0.6 d. 3.5 Number Corner The Math Learning Center