Student Success Adapted Program Lesson-by-lesson support for struggling students!

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1 Student Success Adapted Program Lesson-by-lesson support for struggling students! Lesson Sampler Chapter 2: Numeration

2 Representing Numbers Student Book pp Teacher s Resource pp GOAL Represent numbers to one million using a place value chart, numerals, and words. Preparation and Planning Masters Key Question 2 Assessment Question Checking and Practising BLM 2.1 pp Learning BLM 2.1 pp Place Value Chart to Hundred Thousands, MB p. 43 About the Math In this lesson, students extend their understanding of the place value system up to 1 million. They represent whole numbers up to 1 million using models, symbols, and words. Place value charts and counters are used to represent numbers and to develop and extend an understanding of the place value system. Place value charts, where it is the position of the counters on the chart that determines their values, are a transition step from base ten blocks, where placement is helpful to interpreting the number but does not affect the value. Differentiating Instruction Remind students that place value is simply a way of saying that the position of a digit tells us its value; for example, the 2 in 23 means 20, but the 2 in 42 means 2. Have students concentrate on patterns within the place value system to make its extension more meaningful; that is, we don t have digits greater than 9, so, as soon as we have 10 ones, we trade them for 1 ten in the place to the left of the ones. Similarly, 10 tens make 100, and 10 hundreds make 1000, so it is logical that 10 thousands is the next place value, and 10 ten thousands the next. Some students may need practice modelling numbers to 1000, using base ten blocks or a place value chart to 1000 and counters. You may want to review the words for numbers to 100, particularly the teens. For students who need scaffolding during independent practice, provide Checking and Practising BLM 2.1. For students who need extra learning support, provide Learning BLM 2.1. Answers to Learning BLM (8 tens); 18(8 ones); 851 (8 hundreds) 10; 10; 10 eight; one hundred thousand, , , Thousands eight, ninety-two; eight, fifty-one; eight, ninety-two, eight, fifty-one Reflecting: The line that divides the thousands and ones shows the space; In standard form, the place tells the value. We need 0 to hold the place so that the other digits are in the correct places. Copyright 2009 by Nelson Education Ltd. Overview 2.1: Representing Numbers 37

3 C&P Name: Date: 2.1 Representing Numbers Page 1 Student Book pages Checking 1. a) Model the number on a place value chart. You will need a place value chart counters Look at the ones first The ones digit is 0, so there will be counters in the ones column. There will be counters in the tens column. There will be counters in the hundreds column. Then, look at the thousands There will be counters in the one thousands column. There will be counters in the ten thousands column. Will there be any counters in the hundred thousands column? Sketch your model on the chart below. Thousands b) Write in words. Write 600 in words. Write the word for 4 tens or 40 ones. Put these together to write the number in words. Hint: Do not use the word and between thousands, hundreds, tens, or ones. 38 Checking & Practising BLM 2.1: Representing Numbers Copyright 2009 by Nelson Education Ltd.

4 C&P Name: Date: 2.1 Representing Numbers Page 2 Practising 2. Model each number using counters on a place value chart. Sketch your models on the charts below. a) four hundred seventeen thousand twenty-five Thousands Step 1: Model the 417 thousands. Hint: 17 thousands is 1 ten thousand and 7 one thousands. Step 2: Model the 25 ones. Hint: There are no hundreds in this number. Step 3: Write the number in standard form. Hint: Did you remember to write a 0 in the hundreds place? b) six hundred ninety-one thousand six hundred fifty-nine Thousands Write the number in standard form. Hint: Did you remember to leave a space between the ones and the thousands? Copyright 2009 by Nelson Education Ltd. Checking & Practising BLM 2.1: Representing Numbers 39

5 L Name: Date: 2.1 Representing Numbers Page 1 Student Book pages GOAL Represent numbers to one million using a place value chart, numerals, and words. You will need a place value chart counters In 1997, David Huxley pulled an airplane with a mass of about kg. In 1999, Juraj Barbaric pulled a train with a mass of about kg. How can you model, read, and write these masses? Circle the value of 8 in each number hundreds 8 tens 8 ones 18 8 hundreds 8 tens 8 ones hundreds 8 tens 8 ones Show how the place values are related. 1 ten ones 1 hundred tens 1 thousand hundreds Here is a model of 187 (one hundred eighty-seven). Here is a model of (one hundred eighty-seven thousand). Thousands The three 0s in show that there are no hundreds, tens, or ones. The 0s fill those places so that the 1, 8, and 7 have the correct values. The value of the 7 is seven thousands or The value of the 8 is ten thousands or The value of the 1 is or. 40 Learning BLM 2.1: Representing Numbers Copyright 2009 by Nelson Education Ltd.

6 L Name: Date: 2.1 Representing Numbers Page 2 Write the digits of numerals in groups of 3. space space Start at the right. Leave a space between each group of is 1 million digits 3 digits Here are some other examples: Rewrite the numerals below with the correct space between groups of This way of writing numbers is called standard form. Model on a place value chart. Sketch your model. Thousands standard form The usual way that numbers are written For example, is in standard form. Write in words. hundred thousand Write 851 in words. hundred Write in words. hundred thousand hundred Reflecting How is the space in shown on the place value chart? Why do we need 0 to represent some numbers in standard form? Copyright 2009 by Nelson Education Ltd. Learning BLM 2.1: Representing Numbers 41

7 Using Expanded Form Student Book pp Teacher s Resource pp GOAL Represent, describe, and compare numbers to one million. Preparation and Planning Masters Key Question 2 Assessment Question Checking and Practising BLM 2.2 pp Learning BLM 2.2 pp Place Value Chart to Hundred Thousands, MB p. 43 About the Math In this lesson, students write numbers to 1 million in standard and expanded form. They connect the representation of a number on a place value chart and in expanded form. Students also compare numbers using standard and expanded form. Note that kilometres are introduced in this lesson. Differentiating Instruction Students have learned that a number with more digits is greater than a number with fewer digits, up to Explain that this also applies to greater whole numbers. Ensure that students understand that they should compare the highest place value first, as it has the greatest value. Demonstrate with examples the importance of lining up the digits correctly when comparing numbers; for example, have students compare and Students will also know that a number with more tens is greater than a number with fewer tens. Explain that the same is true when comparing numbers in the thousands. For example, to compare km with km, say, 3 hundred thousands 2 hundred thousands, so Explain that if the value in the highest place is the same, they should compare the value to the right. Have students compare and , then and , and then and Help students connect the representation of a number on a place value chart and in expanded form. You may want to begin with 4-digit numbers and build to 6-digit numbers. As an intermediate step, you may want to have students place number cards (0 9) on a place value chart to represent the number. Provide practice by having students play this game in 2 teams. Place 2 shuffled sets of 0 9 number cards face down. On the chalkboard, draw 2 place value charts to the hundred thousands, 1 for each team. Teams take turns drawing the top card from their deck and placing the number on the place value chart. The object of the game is to make the greatest number, so students will need to be strategic about where they place each number. Once two 6-digit numbers are formed, have students compare the numbers to determine which is greater. For students who need scaffolding during independent practice, provide Checking and Practising BLM 2.2. For students who need extra learning support, provide Learning BLM 2.2. Answers to Learning BLM 2.2 Step 3: 48, 84, 44 Step 4: , , Step 5: 4, 8, 4; , 8000, 400 8, 4, 4; , , 4000, ; ; Reflecting: You don t need to include the 0s in expanded form because they don t have any value. 42 Overview 2.2: Using Expanded Form Copyright 2009 by Nelson Education Ltd.

8 C&P Name: Date: 2.2 Using Expanded Form Page 1 Student Book pages Checking 1. a) It takes the planet Jupiter one hundred three thousand You will need a place value chart counters nine hundred forty-four hours to travel around the Sun. Show this number on the place value chart below. Thousands Write the number in standard form. It takes Saturn hours to travel around the Sun. Show this number on the place value chart below. Thousands Write the number in standard form. b) Does Jupiter or Saturn take longer to go around the Sun? How do you know? Copyright 2009 by Nelson Education Ltd. Checking & Practising BLM 2.2: Using Expanded Form 43

9 C&P Name: Date: 2.2 Using Expanded Form Page 2 Practising 2. Use 8 counters to model 3 different 6-digit numbers. Draw your models in the place value charts below. Model a 6-digit number. Write it in standard form. Thousands Write it in expanded form. Model another 6-digit number. Thousands Write it in standard form. Write it in expanded form. Model another 6-digit number. Thousands Write it in standard form. Write it in expanded form. 44 Checking & Practising BLM 2.2: Using Expanded Form Copyright 2009 by Nelson Education Ltd.

10 L Name: Date: 2.2 Using Expanded Form Page 1 Student Book pages GOAL Represent, describe, and compare numbers to one million. You will need a place value chart counters The distance from Earth to the Moon, in kilometres, is the greatest number you can make with the digits shown, starting with the What is the distance from Earth to the Moon? There are 6 number cards, so the distance is a 6-digit number. Step 1: The problem says that the first digit is 3. Write the first digit in the first column. Thousands 3 Step 2: To make the greatest number, put the 0s in the places with the least value. Note: 0 is smaller than all the other numbers. Write the two 0s in the tens and ones places. Thousands Step 3: Write the 3 different numbers you can make with 4, 4, and Copyright 2009 by Nelson Education Ltd. Learning BLM 2.2: Using Expanded Form 45

11 L Name: Date: 2.2 Using Expanded Form Page 2 Step 4: Write the 3 possible distances. Thousands Step 5: , , and are possible distances. Find out which is greater. Write each number in expanded form. expanded form A way to write a number that shows the value of each digit hundred thousands 4 ten thousands 4 thousands 8 hundreds hundred thousands ten thousands thousands hundreds hundred thousands ten thousands thousands hundreds The number of hundred thousands (3) is the same in each number. Compare the ten thousands. Circle the number that has the most ten thousands So, is the greatest number. The distance between Earth and the Moon is km. Reflecting is a 6-digit number. Why does the expanded form of only have 4 values added together? 46 Learning BLM 2.2: Using Expanded Form Copyright 2009 by Nelson Education Ltd.

12 Renaming Numbers Student Book p. 48 Teacher s Resource pp GOAL Rename numbers that have up to seven digits. Preparation and Planning Masters Explore BLM-A 2.3 p. 48 Explore BLM-B 2.3 p. 49 Key Assessment Question Entire exploration About the Math In this exploration, students rename a 6-digit number using a variety of strategies of their choice, which may include writing the number in expanded form and/or using a place value chart. Some students may interpret the problem as simply asking them to relate the sizes of the containers to the place values, and will generate one solution to the problem. Others may realize that there are a large number of possible ways to fill the containers with the popcorn. Encourage each student to share his or her personal strategy. Litres are introduced in this lesson. Litres and capacity are worked with in greater depth in Lesson 8.8. Differentiating Instruction If students are having difficulty getting started, ask them what they would multiply the capacity of container Z by to get the capacity of container Y (10), and so on. Ask them what the pattern is. (multiply by 10 each time) Have them compare this pattern to the pattern in the place values. As students represent numbers, encourage them to model the numbers in a variety of ways; for example, as well as representing 43 as 4 tens and 3 ones, you can represent it as 3 tens and 12 ones or 43 ones. Help students extend this understanding; for example, is 120 thousands as well as 1 hundred thousand and 2 ten thousands. If students focus on using the standard expansion of the number and do not consider alternate forms, ask them to model 321 using counters and a place value chart (or base ten blocks), and then write the number in expanded form. (3 hundreds 2 tens 1 one) Then, ask students to trade some of the blocks to rename 321. ( for example, 32 tens 1 one, or 3 hundreds 21 ones) Discuss why all the models represent 321. For students who need scaffolding of the exploration, provide Explore BLM-A 2.3. For students working in a lower number range, provide Explore BLM-B 2.3, which presents an adapted version of the central problem for the lesson. Note: The lesson goal has been changed for this BLM. Answers to Explore BLM-B 2.3 A. 1000, 900, 40, 7 1, 9, 4, 7 B. 21, 947 Copyright 2009 by Nelson Education Ltd. Overview 2.3: Renaming Numbers 47

13 E-A Name: Date: 2.3 Renaming Numbers Student Book page 48 A group of students broke a world record by filling a container with L of popcorn. U L V L W 1000 L X 100 L Y 10 L Z 1 L Which of these containers could you use to measure L? Write in expanded form. Use the expanded form to show one way you could measure L L L 1000 L 100 L 10 L 1 L Or, you could fill the 1 L container times to measure L. Look at the two groups of 3 digits in thousands ones You could fill the 1000 L container times and the 1 L container times. Describe other ways you could use the containers to measure L. Idea: Represent using counters on a place value chart. Then, regroup tens as ones, or hundreds as thousands, and so on, to find new combinations. 48 Explore BLM-A 2.3: Renaming Numbers Copyright 2009 by Nelson Education Ltd.

14 E-B Name: Date: 2.3 Renaming Numbers Student Book page 48 GOAL Rename numbers that have up to five digits. You will need a place value chart counters A litre (L) is a unit used to measure capacity. Capacity is the amount that a container can hold. A group of students broke a school record by filling a container with L of popcorn. Which containers could you fill if you had L of popcorn? A. Decide how many of each container you need to make L. Fill in the number of each container in the place value chart below. Measuring L (ten thousands) (one thousands) (hundreds) (tens) (ones) V L W 1000 L X 100 L Y 10 L Z 1 L L 1000 L 100 L 10 L 1 L B. Here is another way to measure L is 21 thousands and 947 ones. You could fill the 1000 L container times and the 1 L container times. Copyright 2009 by Nelson Education Ltd. Explore BLM-B 2.3: Renaming Numbers 49

15 Rounding Numbers Student Book pp Teacher s Resource pp GOAL Round numbers to the nearest hundred thousand, the nearest ten thousand, and the nearest thousand. Preparation and Planning Masters Key Question 3 Assessment Question Checking and Practising BLM 2.4 pp Learning BLM 2.4 pp About the Math In this lesson, students round numbers using place values and number lines. The context of the lesson is population, a context where benchmark numbers are often used. Number lines are particularly suitable for estimating numbers by relating them to benchmark numbers. Students use number lines marked at multiples of , , and 1000 to round numbers to the nearest hundred thousand, ten thousand, and thousand. Differentiating Instruction Ask students to create number lines that can be used to locate the positions of numbers to 10, then to 100, and then to Engage students in a conversation about how the benchmarks they used are similar in each case, so that they can extend those notions beyond Then ask students to create number lines that can be used to locate the positions of numbers to , to , and to 1 million. Ensure that students realize that the same benchmark number can be used to estimate different numbers. Discuss with students why, when a population is given, it is usually not an exact number. Speak deliberately about the difference between estimating and counting exactly; students require explicit exposure to situations where an estimate is all that is required. Other examples are animal populations, distances in astronomy, or large amounts of money. Encourage students to think of a number such as as greater than but less than For those students who need it, provide practice rounding 4-digit numbers to the nearest hundred and thousand; for example, ask students to compare 3278 to 3000 and 4000, and then to 3200 and Ask them to explain how they know which of these numbers is greater or less than Check that they are using place values to compare numbers. For students who need scaffolding during independent practice, provide Checking and Practising BLM 2.4. For students who need extra learning support, provide Learning BLM 2.4, which presents an adapted version of the central problem for the lesson. Note: The lesson goal has been changed for this BLM. Answers to Learning BLM 2.4 A is closer to 9000 on the number line. B C Reflecting: Down; It is closer to the left end of the number line; Up; There are 9 tens, which is almost another hundred; Down; There are 2 ones, and 2 is closer to 0 than 10; 9000; It gives a useful estimate of the size. It shows that not many people live in the Inuvik Region. The population will change, but it will probably stay close to 9000; Overview 2.4: Rounding Numbers Copyright 2009 by Nelson Education Ltd.

16 C&P Name: Date: 2.4 Rounding Numbers Page 1 Student Book pages Checking 1. a)round the populations of Kelowna and Abbotsford to the nearest hundred thousand, the nearest ten thousand, and the nearest thousand. Use number lines. Record your estimates in the table below. Population Nearest Nearest Nearest City in Kelowna Abbotsford Use this number line to round the populations to the nearest hundred thousand. Use this number line to round the populations to the nearest ten thousand. Use this number line to round the populations to the nearest thousand b) Do the 2 populations round to the same hundred thousand? Do the 2 populations round to the same ten thousand? Do the 2 populations round to the same thousand? Copyright 2009 by Nelson Education Ltd. Checking & Practising BLM 2.4: Rounding Numbers 51

17 C&P Name: Date: 2.4 Rounding Numbers Page 2 Practising 3. Vasco looked up the total land and fresh water area of each Western province. a) Round each area to the nearest hundred thousand, ten thousand, and thousand. Use the number lines on this page to help you estimate. Land and fresh water Nearest Nearest Nearest Province (km 2 ) British Columbia Alberta Saskatchewan Manitoba b) Which provinces have the same area, to the nearest hundred thousand? c) Which provinces have the same area, to the nearest ten thousand? Checking & Practising BLM 2.4: Rounding Numbers Copyright 2009 by Nelson Education Ltd.

18 L Name: Date: 2.4 Rounding Numbers Page 1 Student Book pages GOAL Round numbers to the nearest thousand, the nearest hundred, and the nearest ten. The population of an area is the total number of people who live there. The Inuvik Region in the Northwest Territories had a population of 9192 in What other ways can you use to show the population of the Inuvik Region? A. The population of an area is always changing. Yukon Territory Beaufort Sea Port Radium Northwest Nunavut Territories Yellowknife Great Slave Lake Lake Athabasca Prince George Tuktoyaktuk Inuvik Great Bear Lake C A N A D A Hudson Bay You could use an estimate to show the population of the Inuvik Region. Spokane Edmonton Calgary Regina Winnipeg Lake Winnipeg You could round 9192 to the nearest thousand. How do you know that 9000 is the nearest thousand? B. Round the population of the Inuvik Region to the nearest hundred. Hint: Is 9192 closest to 9100, 9200, 9300,...? The population of the Inuvik Region is. C. Round the population of the Inuvik Region to the nearest ten The population of the Inuvik Region is. Copyright 2009 by Nelson Education Ltd. Learning BLM 2.4: Rounding Numbers 53

19 L Name: Date: 2.4 Rounding Numbers Page 2 Reflecting You rounded 9192 to the nearest thousand. Was the number rounded up or down? Why? You rounded 9192 to the nearest hundred. Was the number rounded up or down? Why? You rounded 9192 to the nearest ten. Was the number rounded up or down? Why? Which rounded number would you use to describe the population of the Inuvik Region? Why? If the population of the Inuvik Region increased by 900, what number would you round the population to? 54 Learning BLM 2.4: Rounding Numbers Copyright 2009 by Nelson Education Ltd.

20 Exploring One Million Student Book p. 53 Teacher s Resource pp GOAL Describe one million in various ways. Preparation and Planning Masters Explore BLM-A 2.5 p. 56 Explore BLM-B 2.5 p. 57 Key Assessment Question Entire exploration About the Math In this exploration, students are asked to describe 1 million by comparing 1 million of one thing with a smaller number of something else. Students will draw on their understanding of the metric system to create interesting facts about 1 million. They will likely need to multiply by 1 million and/or divide 1 million by another number. A calculator can be used for such calculations. This exploration will be richer if students are able to use the Internet for research, but students can also use facts found in books and magazines. Differentiating Instruction After reading the book How Much Is a Million? by David M. Schwartz, provide students with a concrete representation of 1 million. Show 1 grain of salt and then 10 grains of salt. Make a tiny pile of salt and tell students that this is about 100 grains. Show a pinch of salt (about 1000 grains); 1 teaspoon (about grains; 1 tablespoon (about grains); and 1 cup (about grains). 2 Ask students what they would name the place value to the left of the hundred thousands place (they might say a thousand thousands, which is an excellent response). Tell them that 1 million is 1000 thousands. Work through a couple of examples with students to demonstrate how to manipulate numbers to create facts involving 1 million. For example, a raisin is about 1 cm long, so 1 million raisins will be about cm long. Then, students can convert this length into one that is easier for them to visualize. There are 100 cm in 1 m, so 1 million cm m. (Have students count to check that there are six 0s after the 1 when they key 1 million into their calculators.) There are 1000 m in 1 km, so m km. Suppose 10 km is the distance from the school to some landmark. Then, a row of 1 million raisins will stretch from the school to that landmark. Or, if a student lives about 1 km from the school, he or she could say that 1 million raisins would make a row stretching from home to school 10 times. Help students see that a number can have a continuous context, such as the distance between 2 places, or a discrete context, such as a number of raisins. Encourage students to research topics they are interested in. For students who need assistance in getting started on the exploration, provide Explore BLM-A 2.5. For students who need extra learning support, provide Explore BLM-B 2.5, which guides students to create a number of interesting facts about 1 million. Answers to Explore BLM-B , 10; , ; ; , Vancouver Copyright 2009 by Nelson Education Ltd. Overview 2.5: Exploring One Million 55

21 E-A Name: Date: 2.5 Exploring One Million Student Book page 53 If you wrote a book about 1 million, what interesting facts would you include? You will need a calculator A humpback whale has a mass of kg humpback whales have a mass of 1 million kg. A Pacific walrus has a mass of 4000 kg. Pacific walruses have a mass of 1 million kg. A bottlenosed dolphin has a mass of 400 kg. bottlenosed dolphins have a mass of 1 million kg. Research another animal mass. Use the mass to write an interesting fact about 1 million. The distance between Vancouver and Winnipeg is about 2000 km. Divide 1 million km by the distance. Use a calculator km km You would need to travel between Vancouver and Winnipeg times to travel 1 million km. Research the distance between 2 other cities in Canada. Use the distance to write an interesting fact about 1 million. 56 Explore BLM-A 2.5: Exploring One Million Copyright 2009 by Nelson Education Ltd.

22 E-B Name: Date: 2.5 Exploring One Million Student Book page 53 GOAL Describe one million in various ways. You will need a calculator If you wrote a book about 1 million, what interesting facts would you include? The mass of a blue whale is about kg. What number can you multiply by to get 1 million? So, blue whales have a total mass of 1 million kg. Cats sleep about 10 hours a day. How many cats will it take to sleep 1 million hours in 1 day? about cats It will take about cats to sleep 1 million hours in 1 day. The distance from Calgary to Vancouver is about 1000 km. 1 km 1000 m Write the distance from Calgary to Vancouver in metres km 1000 m m Length of 1 bobcat 1 m Length of a row of 1 million bobcats m m A row of 1 million bobcats would stretch from Calgary to. Copyright 2009 by Nelson Education Ltd. Explore BLM-B 2.5: Exploring One Million 57

23 Decimal Place Value Student Book pp Teacher s Resource pp GOAL Read, write, and model decimals. Preparation and Planning Masters Key Question 2 Assessment Question Checking and Practising BLM 2.6 pp Learning BLM 2.6 pp Decimal Place Value Chart, MB p. 44 Base Ten Blocks, MB pp About the Math Base ten materials help students understand our base ten place value system. In this lesson, students use base ten blocks on a place value chart to model decimals to thousandths. They write decimals in standard form, expanded form, and words. They also attach a numeral to a model for decimal thousandths. Differentiating Instruction The understanding of decimals is dependent on a previous understanding of the concept of a fraction. Review with students how, in a fraction, the denominator tells the number of parts the whole is divided up into, and the numerator tells how many of those parts are under consideration. Discuss with students how the numerator and denominator of 1, 1, and 1 are represented in the base ten model. To show thousandths, a whole divided into 1000 equal parts is needed. The large cube fits this description. 1 small cube is 1 because it 1000 takes 1000 of these to make 1 whole (the large cube). Similarly, it takes 100 rods to make 1 large cube, so 1 rod is 1, and it takes 10 flats to make 1 large cube, so 1 flat is Model 393 thousandths on a place value chart and ask students to identify the value of each place. (The 3 flats are 3 tenths, the 9 rods are 9 hundredths, and the 3 small cubes are 3 thousandths.) Then, guide students to make the connection that the decimal means 3 tenths, 9 hundredths, and 3 thousandths. Tell students that, in the model they are using, 10 large cubes would represent the whole number 10. Ask students to imagine 10 large cubes, and then 100 large cubes to help them understand just how small those values to the right of the decimal point actually are. Ask students how many small cubes you would use to represent (3 flats is 300 small cubes, 9 rods is 90, and 3 more is 393.) Explain that when they read the decimal or write it in words, they should think of it in small cubes, so they would say 393 thousandths and write three hundred and ninety-three thousandths. For students who need scaffolding during independent practice, provide Checking and Practising BLM 2.6. For students who need extra learning support, provide Learning BLM 2.6, which provides a more structured introduction to the concepts. Answers to Learning BLM , 0.3; 100, 0.25; 1000, , 8, 3, 4; 2, 6, 9, 6 0.1, 0.01, , 9, 3; ; ; three hundred and ninety-three Reflecting: 3 tenths; 3 thousandths; the 3 in the first place after the decimal point 58 Overview 2.6: Decimal Place Value Copyright 2009 by Nelson Education Ltd.

24 C&P Name: Date: 2.6 Decimal Place Value Page 1 Student Book pages Checking 1. a)rachel bought a kg package of trail mix. Model kg using base ten blocks. Hint: Leave the column empty when the place value is 0. Sketch your model. You will need base ten blocks a decimal place value chart Tenths Hundredths Thousandths Write in expanded form. Hint: Do not include place values that are whole hundredths thousandths or or Write kg in words. one and thousandths of a kilogram b) Lauren bought a kg package of trail mix. Model kg using base ten blocks. Sketch your model. Tenths Hundredths Thousandths Copyright 2009 by Nelson Education Ltd. Checking & Practising BLM 2.6: Decimal Place Value 59

25 C&P Name: Date: 2.6 Decimal Place Value Page 2 Write in expanded form whole tenths thousandth or or Write kg in words. Practising 2. A Canadian penny costs cents to make. a) Model on a place value chart. Hint: Leave the column empty when the place value is 0. Sketch your model. Tenths Hundredths Thousandths b) Write the cost in expanded form. Hint: Only include place values that are not 0. The expanded form of is just or or 0.. c) Write the cost in words. 3. a) Write in standard form.. b) Write in standard form.. 60 Checking & Practising BLM 2.6: Decimal Place Value Copyright 2009 by Nelson Education Ltd.

26 L Name: Date: 2.6 Decimal Place Value Page 1 Student Book pages GOAL Read, write, and model decimals. Mateo bought a package of trail mix to take on a hike. The mass of the trail mix is kg. You will need base ten blocks a decimal place value chart How can Mateo model the mass of the trail mix? You can write fractions as decimals. Fraction Decimal 1 tenth hundredth thousandth Complete the chart. Fraction Decimal 3 tenths hundredths thousandths 0. You can write decimals in expanded form whole 2 tenths 5 hundredths 7 thousandths Write these decimals in expanded form whole tenths hundredths thousandths wholes tenths hundredths thousandths You can use base ten blocks to model fractions or decimals. one 1 or 1.0 one tenth 1 or one hundredth or 0. one thousandth or Copyright 2009 by Nelson Education Ltd. Learning BLM 2.6: Decimal Place Value 61

27 L Name: Date: 2.6 Decimal Place Value Page 2 Use base ten blocks to model on a decimal place value chart. Make a copy of this model. Tenths Hundredths Thousandths Write in expanded form whole tenths hundredths thousandths or or small cubes thousandths Write in words. one and thousandths Reflecting In 1.393, the 3 in the 1st place after the decimal point represents. The 3 in the 3rd place after the decimal point represents. Which of these 3s represents a greater mass? 62 Learning BLM 2.6: Decimal Place Value Copyright 2009 by Nelson Education Ltd.

28 Renaming Decimals Student Book pp Teacher s Resource pp GOAL Represent decimals and relate them to fractions. Preparation and Planning Masters Key Question 2 Assessment Question Checking and Practising BLM 2.7 pp Learning BLM 2.7 pp Hundredths Grid, MB p. 37 Thousandths Grid, MB p. 38 Differentiating Instruction About the Math In this lesson, students use hundredths and thousandths grids to model decimals pictorially. They make connections between decimals and fractions, and learn that there are numerous names for the same decimal number. A key concept is that the whole grid is 1 whole, just as the large base ten cube was 1 whole in the model used in Lesson 2.1. It follows that, if 2 grids of the same size are used to represent 2 decimals or fractions, and the same amount of the grid is coloured for each, the 2 numbers have the same value. Although they have the same value, they are different because they cannot always be modelled on the same grid. For example, you can model 0.4 but not 0.40 on a tenths grid. Students need a solid understanding of the relationships between the units on the grids to be able to efficiently use grids to represent numbers and to identify numbers represented on grids. Post a chart of these various relationships that students can refer to. Provide examples to help students better understand the concept Hundredths and Thousandths Grids of a placeholder; for instance, write 0.67 on the chalkboard and ask students to model 67 hundredths with counters on a decimal 1 tenth place value chart. Then, write on the board and ask students to model 1 column 10 hundredths thousandths below the first model on the same chart. Ask what is different about 100 thousandths the models. (nothing) Ask what is different about the decimals. (One has a 0 in the 1 hundredth thousandths place.) Tell students that these 1 square 100 numbers are equivalent because they 10 thousandths represent the same value. 1 For students who need scaffolding during 1 rectangle 1 thousandth independent practice, provide Checking and Practising BLM 2.7. For students who need extra learning support, provide Learning BLM 2.7, which presents the concepts more incrementally. Answers to Learning BLM 2.7 Step 2: 10; 10 Step 3: 0.24 Step 4: 2, 4 Step 5: Step 6: 100; hundredth, 1, 0.01; 10; ; 100 Step 7: Step 8: 2, 4, 0 Reflecting: 24 hundredths 2 tenths 4 hundredths, and 240 thousandths 2 tenths 4 hundredths 0 thousandths. The 0 thousandths is equal to 0, so the two decimals are the same. Copyright 2009 by Nelson Education Ltd. Overview 2.7: Renaming Decimals 63

29 C&P Name: Date: 2.7 Renaming Decimals Page 1 Student Book pages Checking 1. There are 1000 students at Belle s school. 400 students play an instrument. You will need pencil crayons thousandths grids a) Colour a thousandths grid to show 400 out of 1000 students. Hint: Each column is 1 tenth , so each column is thousandths. b) Write a fraction to represent the coloured part of the grid. Write this fraction as a decimal Each square on the grid is 1. Count the number of squares you coloured. Write another fraction to represent the coloured part of the grid. Each column on the grid is 1. Count the number of columns you coloured. Write another fraction to represent the coloured part of the grid. Write this fraction as a decimal Checking & Practising BLM 2.7: Renaming Decimals Copyright 2009 by Nelson Education Ltd.

30 C&P Name: Date: 2.7 Renaming Decimals Page 2 Practising 2. Emanuel coloured part of a thousandths grid. a) Write a fraction to represent the coloured part. b) Write a decimal thousandth to represent the coloured part. 5. a) 0.29 Write the decimal in expanded form. tenths hundredths 1 column is 1 tenth. 1 square is 1. Colour the decimal on the grid. Write the decimal as an equivalent decimal thousandth b) 0.68 Colour the decimal on the grid. Write the decimal as an equivalent decimal thousandth Copyright 2009 by Nelson Education Ltd. Checking & Practising BLM 2.7: Renaming Decimals 65

31 L Name: Date: 2.7 Renaming Decimals Page 1 Student Book pages GOAL Represent decimals and relate them to fractions. Anne goes to a school with 100 students. Belle goes to a school with 1000 students. There are 24 Grade 5 students in Anne s school. There are 240 Grade 5 students in Belle s school You will need pencil crayons thousandths grids hundredths grids How can you use decimals to compare the Grade 5 students in the 2 schools? Step 1: Write a fraction for the number of Grade 5 students in Anne s school Step 2: Model the fraction on a hundredths grid. There are 10 columns in a hundredths grid. Each column is one tenth or 1 or There are 100 squares in a hundredths grid. Each square is one hundredth or 1 or How many squares are in 1 column? So, one tenth hundredths. Colour 24 hundredths on the grid. Step 3: Write the fraction 24 as a decimal Step 4: Write the fraction 24 in expanded form is the same as 24 hundredths hundredths is tenths hundredths. 66 Learning BLM 2.7: Renaming Decimals Copyright 2009 by Nelson Education Ltd.

32 L Name: Date: 2.7 Renaming Decimals Page 2 Step 5: Write a fraction for the number of Grade 5 students in Belle s school. Step 6: Model the fraction on a thousandths grid. There are 10 columns in a thousandths grid. Each column is one tenth or 1 or There are squares in a thousandths grid Each square is one or or 0.. How many squares are in a column? So, 1 tenth hundredths. There are 1000 rectangles in a thousandths grid. Each rectangle is one thousandth or 1 or There are 100 rectangles in a column. So, 1 tenth thousandths. Colour 240 hundredths on the grid. Step 7: Write the fraction Step 8: Write the fraction is the same as 240 thousandths. as a decimal. 0. in expanded form. 240 thousands is tenths hundredths thousandths The amount that is coloured on both grids is the same. The decimals 0.24 and are equivalent decimals. Reflecting equivalent Having the same value For example, How did writing both decimals in expanded form show that they are equivalent? Copyright 2009 by Nelson Education Ltd. Learning BLM 2.7: Renaming Decimals 67

33 Communicating about Equivalent Decimals Student Book pp Teacher s Resource pp GOAL Explain whether two decimals are equivalent. Preparation and Planning Masters Key Question 2 Assessment Question Checking and Practising BLM 2.8 pp Learning BLM 2.8 pp Tenths Grid, MB p. 36 Hundredths Grid, MB p. 37 Thousandths Grid, MB p. 38 About the Math In this lesson, students communicate their understanding of equivalent fractions and decimals. Encourage a variety of strategies and have materials (place value charts, thousandths and hundredths grids, counters, base ten blocks) available for those who wish to use them. Review and post the Communication Checklist to help remind students of the expectations. Differentiating Instruction Have students brainstorm math words to use in their explanations. Post a chart of these words: decimal, fraction, expanded form, equivalent, tenth, hundredth, thousandth, place value chart, hundredths grid, thousandths grid, model, represent, column, square, rectangle, regroup, rename, and so on. Have students in pairs describe the decimal 0.6. Then, ask students to share their description with the rest of the class, and record their answers on chart paper. (It is a decimal; There are 6 tenths; There is no whole number before the decimal.) Or, students may make a model. Next, have students describe Again, have them share their descriptions and record their answers. (There are 60 hundredths; The 6 is in the tenths place; The number is less than 1 because there is no whole number before the decimal.) Or, students may make a model. Ask students if the following statement is true: 0.6 and 0.60 are equivalent decimals. (Yes, because the 2 grids are the same size, and each decimal is represented by the same amount of a grid. On a place value chart they have the same numbers of counters in the same places, etc.) For students who need scaffolding during independent practice, provide Checking and Practising BLM 2.8. For students who need extra learning support, provide Learning BLM 2.8. Answers to Learning BLM 2.8 Stefan used math language (e.g., models, represent, decimals, tenths grid) and included a diagram. He did not include the right amount of detail. He modelled 0.5 on a tenths grid because it is tenths; 0.50 on a hundredths grid because it is hundredths; on a thousandths grid because it is thousandths. The 3 grids are all the same size, so equal amounts coloured have the same value. Reflecting: It was easy to see and compare the values of the decimals. 68 Overview 2.8: Communicating about Equivalent Decimals Copyright 2009 by Nelson Education Ltd.

34 C&P Name: Date: 2.8 Communicating about Equivalent Decimals Page 1 Student Book pages Checking 1. Emily explained why 0.2 and 0.20 are equivalent. I can model 0.2 and 0.20 on a place value chart. They are equivalent. Communication Checklist Did you use math language? Did you include the right amount of detail? Did you include a diagram? Use the Communication Checklist to improve Emily s explanation. Did Emily use math language? Underline the math language Emily used in her explanation. Did Emily include the right amount of detail? Rewrite Emily s explanation using more detail. I can model 0.2 and 0.20 on a place value chart. 0.2 is tenths and 0.20 is hundredths. I can regroup 20 hundredths as 2. So, 0.2 and 0.20 are decimals. Did Emily include a diagram? Show 0.2 and 0.20 on the place value chart below. Tenths Hundredths Thousandths Copyright 2009 by Nelson Education Ltd. Checking & Practising BLM 2.8: Communicating about Equivalent Decimals 69

35 C&P Name: Date: 2.8 Communicating about Equivalent Decimals Page 2 Practising 2. Jeremy and Anna are driving to Peace River with their parents. Jeremy says that they have driven 0.3 of the way. 30 Anna says that they have driven of the way. 100 Explain why they are both right. Represent 0.3 and on the hundredths grids Use your diagrams to help you explain why 0.3 and Use the Communication Checklist are equivalent. Communication Checklist Did you use math language? Did you include the right amount of detail? Did you include a diagram? 70 Checking & Practising BLM 2.8: Communicating about Equivalent Decimals Copyright 2009 by Nelson Education Ltd.

36 L Name: Date: 2.8 Communicating about Equivalent Decimals Page 1 Student Book pages GOAL Explain whether two decimals are equivalent. Stefan has a chocolate bar. His brother, Colin, wants him to share it. Stefan tells Colin that 0.5, 0.50, and of the chocolate bar are the same amount. Colin wants to know why. How can Stefan explain how he knows that the decimals are equivalent? Stefan s Explanation I ll use models to represent the chocolate bar, and I ll colour the decimals. I can model 0.5 on a tenths grid. I can model 0.50 on a hundredths grid. I can model on a thousandths grid. The decimals 0.5, 0.50, and are equivalent because the same amount is coloured on all 3 grids. Copyright 2009 by Nelson Education Ltd. Learning BLM 2.8: Communicating about Equivalent Decimals 71

37 L Name: Date: 2.8 Communicating about Equivalent Decimals Page 2 What did Stefan explain well? Use the Communication Checklist. Communication Checklist Did you use math language? Did you include the right amount of detail? Did you include a diagram? Improve Stefan s explanation. Explain why Stefan modelled 0.5 on a tenths grid, 0.50 on a hundredths grid, and on a thousandths grid. Explain why Stefan could compare the coloured amounts on the 3 different grids. Hint: How are the grids alike? How are they different? Reflecting How did the diagrams help Stefan explain? 72 Learning BLM 2.8: Communicating about Equivalent Decimals Copyright 2009 by Nelson Education Ltd.

38 Rounding Decimals Student Book pp Teacher s Resource pp GOAL Interpret rounded decimals, and round decimals to the nearest tenth or the nearest hundredth. Preparation and Planning Masters Key Question 2 Assessment Question Checking and Practising BLM 2.9 pp Learning BLM 2.9 pp Hundredths Grids, MB p. 37 Thousandths Grids, MB p. 38 About the Math In this lesson, students use decimal representations on thousandths grids to help them round decimals. The lesson provides an example of how meaningful answers can often be obtained through estimation. Before students begin rounding, review the values of the sections on a thousandths grid: columns are tenths, squares are hundredths, and rectangles are thousandths. Encourage students to draw on their experience rounding whole numbers the conventions for rounding decimals exactly parallel those used for whole numbers. Differentiating Instruction Show students represented on a thousandths grid. Ask them to round to the nearest hundredth: How many full squares are coloured? (15) How many rectangles are coloured? (2) Are 2 rectangles close to 1 full square? (no) So, will you add another square to 15 when you round, or will you round to 0.15? (Round to 0.15) Ask students to round 152 to the nearest tenth (150) and compare this to the way they rounded (It s the same, we dropped the 2.) Ask students to round to the nearest tenth: How many full columns are coloured? (1) What does 1 column represent? (one tenth) Besides the 10 squares in the column, how many full squares are coloured? (5) Will you count this half of a column as another tenth or not? Explain. (Yes, because when the number in the place you re rounding off at is 5 or more, you count it as 1 in the next place.) Do you need to consider the thousandths when you re rounding to the nearest tenth? (no) For students who need scaffolding during independent practice, provide Checking and Practising BLM 2.9. For students who need extra learning support, provide Learning BLM 2.9, which presents an adapted version of the central problem for the lesson. Note: The lesson goal has been changed for this BLM. Answers to Learning BLM 2.9 A. less than 3; closer to 3; 3 B. 0.3; 3 C. 30; 30; 30 Reflecting: 7; 1; Since 7 full columns are shaded in and only 1 square is shaded in on the next column, the number is rounded to 7 tenths. Copyright 2009 by Nelson Education Ltd. Overview 2.9: Rounding Decimals 73

39 C&P Name: Date: 2.9 Rounding Decimals Page 1 Student Book pages Checking 1. The chart on this page shows batting averages for 2 professional baseball players. Batting averages are reported in decimal thousandths. A batting average of means the player can expect to get 447 hits in 1000 times at bat. a) Model J. McDonald s batting average of on the thousandths grid provided here. You will need pencil crayons thousandths grids Batting Averages Player Batting average J. McDonald R. Clayton b) Round to the nearest hundredth is about c) J. McDonald will probably get about hits in 100 times at bat. d) Round to the nearest tenth is about e) J. McDonald will probably get about hits in 10 times at bat. f) Round R. Clayton s batting average of without using a grid. Think of 288 as a whole number. You can round 288 to 290. You can round to 0.. You can round 288 to 300. You can round to Checking & Practising BLM 2.9: Rounding Decimals Copyright 2009 by Nelson Education Ltd.

40 C&P Name: Date: 2.9 Rounding Decimals Page 2 Practising 2. Round each decimal to the nearest hundredth and the nearest tenth. Circle the nearest hundredth and nearest tenth for each decimal in the chart below. a) b) c) d) Decimal Nearest hundredth Nearest tenth Which numbers below round to the same hundredth? Look at the digits in the tenths place in each number. Could round to the same hundredth as the other 3 numbers? Explain why or why not. Look at and Would you round to 0.23 or 0.24? Would you round to 0.23 or 0.24? Do these 2 numbers round to the same hundredth? Look at Would you round to 0.22 or 0.23? Which of the other numbers rounds to the same hundredth? Copyright 2009 by Nelson Education Ltd. Checking & Practising BLM 2.9: Rounding Decimals 75