Grade 4 Math Expressions Check Your Understanding Questions 1 Check Your Understanding Questions Unit: Fluency Plan Grade 4 Unit Fluency Plan Fluency Plan Fluency Plan Lesson Activity 1 2 Page Number 2 4 3 9 4 12 1 2 16 18 3 20 4 21 5 23 1 29 Make a drawing to show a 3 by 6 array and then write a multiplication equation to represent the total number of objects. 3 x 6 = 18 2 31 The Commutative Property of Multiplication allows you to rotate arrays in two different ways. Rotate a 3 by 6 array to show a 6 by 3 array. 3 32 Have students solve this problem. Check students’ drawings. The drawing should have 4 rows with 6 stamps in each row. You could arrange 6 rows of 4 stamps; 1 x 24 and 24 x 1; 2 x 12 and 12 x 2; 3 x 8 and 8 x 3. Brendan has a page of stamps. There are 4 rows and 6 stamps in each row. Draw a picture of this page. Can you arrange the stamps in different ways? 1 2 Check Your Understanding What is 9 + 3? 12 How can you use the Associative Property of Addition to add 90 + 20 + 80? Possible answer: First add the 20 + 80 = 100 + 90 = 190. How can you use counting on to subtract 8 from 15? Start at 8 and count on up to 15: 9, 10, 11, 12, 13, 14, 15. I started at 8 and counted on 7 more to get to 15. So 15 – 8 = 7. How is the Commutative and Associative Properties of Addition different? The Commutative Property of Addition involves adding only 2 addends while the Associative Property involves adding 3 or more addends. What is 7 x 8? 56 What three symbols can you use to represent multiplication? • ∗ × List the 2s count-bys up to 20. 2, 4, 6, 8, 10, 12, 14, 16, 18, 20; List the 5s count-bys up to 50. 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 List the 9s count-bys up to 90. 9, 18, 27, 36, 45, 54, 63, 72, 81, 90; List the 10s count-bys up to 100. 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 Explain how you can use Quick Fingers to show 9 x 4. Bend down the 4th finger on the left hand. There are 3 fingers to the left of the bent finger and 6 fingers to the right of the bent finger. This shows 3 tens and 6 ones. So the answer is 36. 3 Grade 4 Math Expressions Check Your Understanding Questions 2 Fluency Plan Fluency Plan Fluency Plan 4 1 2 38 39 N/A N/A 1 44 2 45 3 47 List the 3s count-bys up to 30. 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 How can you use the 5s shortcut to find the product of 8 × 3? Multiply 5 times 3 first, which is 15, and then add three more groups of 3, which is 9. 15 + 9 = 24. Have students solve the problem and identify what type of problem it is. 3 balloons 5 6 4 48 1 52 2 53 3 56 1 60 Complete the statement to make it true. 1; 0 62 When any number is multiplied by _______, the result is that number. When any number is multiplied by _______, the result is 0. Complete the statement to make it true. 1; number 2 Fluency Plan Fluency Plan Fluency Plan List the 4s count-bys up to 40. 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 How can you use the 5s shortcut to find the product of 9 × 3? Multiply 5 times 3 first, which is 15, and then add four more groups of 3, which is 12. 15 + 12 = 27. Copy and complete the Fast 8 Array. 3 24 Dividing a number into that number of equal groups gives each group a size of _______. Dividing a number into 1 group gives each group size equal to the _______. 7 8 The clown brought 24 balloons to the party. He divided the balloons equally among 8 children. How many balloons did each child get? What does the key in a pictograph tell you? The key tells you how many each symbol in the graph represents. 3 64 Write an addition expression and a multiplication expression that equals the same answer. Possible answers: 5 + 0 and 5 x 1 1 2 68 69 N/A N/A 1 74 What are the 8 equations that go with this Fast Array drawing? 4 × 7 = 28 7 × 4= 28 9 2 3 77 78 28 ÷ 4 = 7 28 ÷ 7 = 4 7 4 28 = 7 × 4 7 = 28 ÷ 4 28 28 = 4 × 7 4 = 28 ÷ 7 How can you check your division? use multiplication N/A Grade 4 Math Expressions Check Your Understanding Questions 3 1 Fluency Plan Fluency Plan Fluency Plan 10 82 2 83 3 83 4 84 1 88 2 89 Math Connection 92 1 96 2 99 3 102 11 12 Have students solve this word problem and then tell the type of word problem it is and solve it. 48 roses; Repeated-groups multiplication There are 6 roses in each vase at the flower shop. If there are 8 vases on display, how many roses are there? Have students share their word problems they wrote for problem number 5 on Student Activity Book page 33 and have the class solve them. Check students’ work. Have students share their written word problems and have the class solve them. Check students’ work. What other operation can you use to divide? Repeated subtraction List the 6s count-bys up to 60. 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 How can you combine 2 multiplications you know to multiply 7 x 6? Sample combinations: (4 x 6) + (3 x 6) = 24 + 18 = 42 If the pattern continues in the function table in problem 5 on Student Activity Book page 36, how much money would 9 tickets cost? $72 List the 8s count-bys up to 80. 8, 16, 24, 32, 40, 48, 56, 64, 72, 80 List the 7s count-bys up to 70. 7, 14, 21, 28, 35, 42, 49, 56, 63, 70 Have students solve this problem. 6 butterflies A butterfly movie lasted for 60 minutes. The movie spent 10 minutes on each butterfly. How many different types of butterflies were shown in the movie? 1 2 106 108 N/A Fill in the unknown number in the Factor Triangle. 42 ÷ Fluency Plan 7 13 3 Fluency Plan 6 N/A Write these expressions on the board and have students use greater than and less than symbols to compare the expressions. 46 + 98 > 25 + 37 1 115 2 117 When you multiply a number by the same number (i.e. 7 x 7) what type of an array is formed? A square What is 11 x 9? 99 1 2 3 122 123 124 14 15 × 109 110 Math Connection Fluency Plan ÷ N/A N/A Grade 4 Math Expressions Check Your Understanding Questions 4 Fill in the unknown number in the Factor Triangle. 27 ÷ 9 1 128 ÷ × 3 Give an example of these multiplication properties: Sample answers: Commutative Property of Multiplication: 5 × 7 =7 × 5 2 Fluency Plan 16 3 Fluency Plan 129 17 131 4 132 1 2 138 139 140 Math Connection Identity Property of Multiplication: 5 × 1 =5 Give an example of this multiplication property: Sample answer: Associative Property of Multiplication: (3 × 7) × 2 = 3 × (7 × 2) Give an example of this multiplication property: Sample answers: Distributive Property of Multiplication: 3 × 4 + 3 × 4; 3 (4 × 2) If parentheses are not written in an expression, what operations are done first? Multiplication and division are done before addition and subtraction. N/A N/A Ask students if the symbol could have been 3 cars? 3 cars would not have worked as 3 is not a factor of 4, 20, 16 or 8. It only is the factor of 18. Grade 4 Math Expressions Check Your Understanding Questions 5 Check Your Understanding Questions Unit 1- Solve Multiplication and Division Problems Grade 4 Unit 1 1 1 1 1 Lesson Activity 1 2 Page Number 144 145 A table uses rows and columns to display information. The rows go ______________ and the columns go ____________. 1 2 3 147 Extension 148 1 152 2 154 1 158 3 2 160 1 166 4 5 Check Your Understanding N/A Have students fill in the blanks below to complete the statements. Across/right to left/horizontally; up and down/top to bottom/vertically 2 168 1 172 2 173 3 174 Extension 176 Have students share their questions and table from numbers 6 –7 on Student Activity Book page 59 and have the class answer the questions. Tables, questions, and answers will vary. Look at the function tables on Student Activity Book page 60. What do the equations at the bottom of each function table show? The rule for the function table. If you added another vegetable topping to the table of combinations on Student Activity Book page 63, how many different one-topping pizzas are there now? 12 pizzas What are the different ways you can represent a combination problem? Possible answers: numerically with an equation, a Fast Array drawing, a Tree Diagram, a table, etc. Have students write situation and solution equations for this problem and solve it. Situation equation: 8o = 32; solution equation: o x 8 = 32 or 32 ÷ 8 = o; o = 4 Bali put 32 oranges into 8 equal cartons. How many oranges were in each carton? What is the letter used to represent an unknown in a situation or solution equation called? a variable Have students make comparison statements using multiplication and division about this situation. Sample comparison statements: Tran has 1/3 as many stuffed animals as Renaldo. Renaldo has 3 times as many stuffed animals as Tran. Tran has 3 stuffed animals. Renaldo has 9 stuffed animals. What drawings/tools can you use to help you solve word problems that compare things? Comparison bars What does the key in a pictograph tell you? The key tells you how many each symbol in the graph represents. How is a vertical bar graph arranged? A vertical bar graph has the bars going up and down, not across. What is a survey? Something used to collect information and data. Look at the Heights table on Student Activity Book Grade 4 Math Expressions Check Your Understanding Questions 6 page 72. If another building was 67 stories high, how would you add that piece of data to the stem and leaf plot? You would add a row with a stem of 6 and a leaf of 7. 1 1 6 7 1 181 2 183 Extension 184 1 188 Give examples of hidden information in a problem. Possible answers: Week = 7 days; dozen = 12, etc. 1 2 196 197 N/A Have students fill in the blanks to make these statements correct. prime; composite 3 1 198 8 What does horizontal and vertical mean? Horizontal means “going across” and vertical means “going up and down.” Name all the parts of a bar graph. Title, axes, bars, scale, labels. What does the key on a double bar graph tell you? What each bar stands for. A __________ number is a number greater than 1 that have 1 and itself as the only factors. A _________ number is a number greater than 1 that has more than two factors. Use Factor Fireworks (or a Factor Tree) to show all the factors of the number 16. 16 8 × 2 4 × 2 4 201 1 206 2 207 2 × 2 What are multiples? Multiples are another name for count-bys (i.e. 3, 6, 9, 12). Use the running speed table on Student Activity Book page 83 to answer this question. Pronghorn antelope Which animal almost runs twice as fast as a house cat? Have students fill in the blanks to make these statements complete. Even; odd, even The product of two even whole numbers will always be ___________. The product of two odd whole numbers will always be ___________. The product of an even and an odd whole number will always be ___________. 1 9 3 208 4 208 5 208 What if Cora had 3 boxes instead of 5 boxes, how many muffins would go in each box? 10 muffins in each box Have students point out the inverse operations in this age riddle that “undo” each other. Multiplying and dividing by 2;subtracting and adding 10. Have students solve this word problem. 30 ÷ 7 is 4 R2. 5 pages Edgar plans to write 30 math problems. He can fit 7 math problems on a page. How many pages will Edgar need for all 30 math problems? Grade 4 Math Expressions Check Your Understanding Questions 7 Check Your Understanding Questions Unit 2- Quadrilaterals Grade 4 Unit 2 Lesson Activity 1 Page Number 216 2 218 3 220 4 221 Check Your Understanding Have students draw two congruent triangles on their MathBoard. Triangle shapes will vary, but make sure the two shapes that are drawn are the same shape and the same size. Have students draw two similar triangles on their MathBoard. Triangle shapes will vary, but make sure the two triangles that are drawn are the same shape but may differ in size. Have students draw a triangle with a line of symmetry on their MathBoard. Sample drawing: 1 Math Connection 222 Have students read this statement and fill in the blank. congruent If a figure has a line of symmetry, the two figures formed by the line of symmetry are _____________. Have students read this statement and fill in the blank. reflection The figure on one side of a line of symmetry is a mirror image called a _______________. 1 226 Draw parallel line segments and perpendicular lines. Sample drawings: Parallel line segments 2 227 Perpendicular lines Have students name the line segments, vertex points, and angles in this figure. Line segments: AD, BC; vertex point: E; angles: <BED,<DEC, <AEB, <AEC (angle names may vary). 2 A 2 B E C 3 228 D Draw a triangle. Name it CDE and describe various parts of the triangle. Sample triangle. Vertices: C, D, E; angles: <CDE, etc. C E D Grade 4 Math Expressions Check Your Understanding Questions 8 2 2 2 2 1 232 2 3 233 235 Math Connection 236 1 240 2 242 3 243 4 246 Problem Solving Strategy 248 1 252 2 3 253 255 Extension 256 1 260 2 261 3 263 Math Connection 264 3 4 5 6 Look at Student Activity Book page 99. Which figures have square corners/right angles? A, C, F, H, M True or false, a quadrilateral is a closed figure. True Name as many quadrilaterals as you can. Possible quadrilaterals: square, rectangle, rhombus, kite, diamond, square, parallelogram, trapezoid, isosceles trapezoid What type of a diagram can you use to sort quadrilaterals? A Venn diagram What are you measuring when you measure the perimeter? You are measuring the distance/length around a shape. How can you find the area of a rectangle? Multiply one side by the other. What are the area and perimeter formulas for a square? A= s x s; P = 4 x s What are some different rectangles that have an area of 24 centimeters? What is the perimeter of each? 1 x 24 or 24 x 1 rectangle (P =50 centimeters); 2 x 12 or 12 x 2 rectangle (P =28 centimeters); 3 x 8 or 8 x 3 rectangle (P =22 centimeters); 4 x 6 or 6 x 4 rectangle (P = 20 centimeters); When using the Guess and Check Strategy, if your first number you used gave an answer that was too high, what do you know you have to do to your number? You need to pick a smaller number. How do you find the perimeter of a parallelogram? Possible method: Add the length of the base (b) and the length of the slant height (s). Multiply the total by 2. (b + s) 2 = P How do you find the area of a parallelogram? A= b × h True or false, the height of a parallelogram is parallel to its base. False, the height is perpendicular to the base. What do you need to do to the sides of a rectangle to triple it’s perimeter? Triple both sides Look at the new pool on Student Activity Book page 113. What shapes make up the new, complex figure? 2 rectangles or a rectangle and a square. Have students show the different ways you can decompose figures 10 and 11 on Student Activity Book page 114. Check student’s decompositions. What does the word dimension mean? Dimension means measurement. Have student pairs explain to the class how they calculated the estimates of their feet outlines. Answers will vary. Grade 4 Math Expressions Check Your Understanding Questions 9 Check Your Understanding Questions Unit 3- Place Value Multi-Digit Addition and Subtraction Grade 4 Unit 3 Lesson Activity 1 Page Number 272 2 273 3 274 Check Your Understanding Write 4 + 2 on the left side of the board and 3 + 4 on the right side. Ask, “Are these equal or not equal?” Not equal. 6 ≠ 7 so 4 +2 ≠ 3 +4. Which property of addition states that addition can be done in any order, for example a + b = b + a. Commutative Property of Addition Write 8 related addition and subtraction equations for this break-apart drawing. 1 46 3 3 2 Extension 276 1 280 2 284 1 288 2 289 3 289 1 294 3 46 = 25 + 21 46 = 21 + 25 21 = 46 – 25 25 = 46 – 21 25 + 21 = 46 21 + 25 = 46 46 – 21 = 25 46 – 25 = 21 25 21 How do you show greater than and less than in an inequality? Greater than (>); less than (<) What two types of change situations can you have? Change plus and change minus Have students share their change plus and change minus problems on Student Activity Book page 126 and have the class try and solve them. Problems and answers will vary. What are the three kinds of collection situations? No action, Put Together, and Take Apart. Write these two words on the board and ask students to write an appropriate label for the two words. flowers Rose daisy Have students share their collection problems from number 15 on Student Activity Book page 129 and have the class try and solve them. Problems and answers will vary. Draw these comparison bars on the board and have students state two comparison statements. Statements will vary. Sample statements: Devi has 8 more badges than Ali. Ali has 8 fewer badges than Devi. 3 Devi 4 Ali 2 298 8 Have students share their comparison problems from number 10 on Student Activity Book page 133 and have the class try and solve them. Problems and answers will vary. Grade 4 Math Expressions Check Your Understanding Questions 10 3 1 302 2 304 5 Zoe is a pet sitter. On Tuesday she had 2 dogs, 3 cats, and 2 birds. On Wednesday she watched 3 dogs, 2 cats, 2 fish, and 3 birds. How many more animals did Zoe watch on Wednesday? 1 3 3 3 3 308 Have students fill in the blanks to make this statement complete. Addends; total; groups; size Addition and subtraction problems involve two _______ and a __________. Multiplication and division situations involve repeated __________ of the same ____________. 6 7 True or False, You can combine two steps of a problem in 1 equation. True Have students solve this two-step problem. 3 animals 2 312 Have students use the inverse operations of each problem on Student Activity Book page 138 to check their results. Check students’ work. Students should use subtraction to check addition and vice a versa and multiplication for division and vice a versa. 1 317 Have students show the number 143 on their dot array. 2 3 318 320 How many hundreds are in 1, 000? 10 Have students show the number 1, 228. Math Connection 322 How many hundreds, tens and ones are in $578? 5 hundreds, 7 tens, 8 ones 1 327 2 330 3 332 Represent the number 4, 283 with Secret Code Cards. Check that the 4,000, the 200, the 80, and the 3 Secret Code Cards are assembled correctly. Round the following numbers to the nearest ten: 40, 430, 880, 80 35 425 878 78 Round the following numbers to the nearest hundred: 400, 500, 800, 1, 000 350 470 820 950 1 336 2 337 8 9 Write these numbers on the board and have students use greater than and less than symbols to compare the numbers. Encourage students to use place value drawings to help them. 4, 286 > 4, 238 Write the greatest and least four-digit number possible. Grade 4 Math Expressions Check Your Understanding Questions 11 3 339 Math Connection 340 Use each of the 4 digits in the group exactly once. 7, 643; 3, 467 7346 Write each group of numbers in order from least to greatest. Tell whether the first or last number is closer to the middle number. 36, 63, 78; last number is closer to the middle. 36, 78, 63 Have students fill in the blanks to make the statement true. equal An equality is an equation that indicates two amounts are ___________, and includes an equals (=) sign. 3 3 3 3 1 344 2 348 1 352 2 354 3 356 Math Connection 358 1 2 362 364 Add 67, 897, 654 + 4, 567, 238. 72, 464, 802 When adding numbers, what must you align? The place value places 1 368 2 370 Estimate the total for this equation 628 + 367 = [ ]. 600 + 400 is 1, 000. Add the following numbers. Hint: look for “easy” combinations. 25; some combinations may be 8 and 2, 7 and 3, 2 and 3, etc. 3 372 8+2+7+3+5 Solve this addition problem and then use estimation to check your answer. 7, 463; 5, 000 + 3, 000 = 8, 000 10 11 12 Write this number using words. Thirty-seven thousand, eight hundred ninety-five 37, 895 How many thousands make 1 million? 1, 000 thousands Use any method to add 365 and 678. 1, 043; Methods will vary. Use the New Groups Above, New Groups Below or the Show Subtotals Methods to add 447 + 862. 1, 309; Check students’ methods. Write this equation on the board and have students solve and explain if they need to make new groups? 156 + 77= [ ] 233; Students need to group ones into a new ten, and group tens into a new hundred. Add $53 and $27. $80 13 4, 568 + 2, 895 3 3 1 376 2 378 1 2 382 386 14 15 Why should you draw a magnifying glass around the top number in a subtraction problem? The magnifying glass helps you remember to look at the top number closely and check if you have to do all the ungrouping before you subtract. Estimate the total for this equation 628 – 367 = [ ]. 600 – 400 is 200. What is the inverse operation of addition? subtraction Use addition to check this subtraction problem. Grade 4 Math Expressions Check Your Understanding Questions 12 128 + 72 = 200 9 1 10 10 200 – 72 128 3 16 1 390 2 393 Math Connection 394 In which direction can you ungroup in a subtraction problem? You can either ungroup left to right or right to left. Decide if you need to ungroup in this subtraction exercise. 367 – 158. 209; You need to ungroup 1 ten to get 10 more ones. Have students use a proof drawing to solve this word problem. $29.41; Check students’ proof drawing. Sandy’s neighbour gave her $50.00 for shovelling her driveway all winter. She spent $20.59 on a comic book collection. How much money does she have left? 3 3 3 17 18 1 398 2 401 3 402 1 406 2 407 Math Connection 410 1 414 2 415 3 416 19 When making a Proof Drawing to check your subtraction, do you draw both numbers with boxes, tens, and ones and then subtract? No, you only draw the large number and you take away the smaller number from that drawing. Decide if you need to ungroup in this subtraction exercise. 3,670 – 1,580. 2,090; You need to ungroup a hundred to get 10 tens. Decide if you need to ungroup in this subtraction exercise. 4,660 – 2,340. 232; You don’t need to ungroup at all. Subtract these large numbers. 1, 346, 297; Check students’ ungroupings. 5, 789, 675 – 3, 567, 986 How can you check your subtraction work? Use addition or “adding it up.” Estimate the difference for this equation 6, 428 – 4, 867 = [ ]. 6, 000 – 5, 000 is 1, 000. Have students share their price lists and word problems from problem 5 on Student Activity Book page 171 and have the class solve the problems. Price lists and problems will vary. Look at the total acre column on Student Activity Book page 172 and choose the best rounding unit for each measurement (the nearest ten thousand, the nearest, hundred thousand, or the nearest million). Possible rounded acreages listed from top to bottom: 800,000; 300,000; 50,000; 250,000; 1,500,000; 8,500,000; 1,000,000; 1,000,000; 500,000; 600,000; 50,000; 900,000; 200,000; Discuss different rounding units. Have students share their real-world data and word problems from problem 11 on Student Activity Book page 173 and have the class solve the problems. Data and problems will vary. Grade 4 Math Expressions Check Your Understanding Questions 13 3 3 20 4 417 Problem Solving Strategy 418 1 422 2 424 1 428 2 429 3 430 4 430 5 430 21 When you use a calculator to check results, the calculator does not always show commas. How do you know where to insert the commas? The commas are placed between the hundreds and thousands places, the hundred thousands and millions places, etc. When you estimate, does your answer have to be correct? No, estimating is just trying to make a close guess of what you think the answer or measurement will be. Some answers can be estimates (i.e. time, money, distances, etc.) while others need to be exact. What are the three types of problems that involve addition and subtraction? Change plus/minus; collection; comparison List as many ways you can check you work/ computation on Student Activity Book page 176. Possible methods: estimation, calculator, inverse operations, proof drawing, etc. Have students share their question from problem 3 on Student Activity Book page 177 and have the class answer it using the data table. Questions and answers will vary. Have students show their tally charts and bar graphs from their surveys. Have the class make as many statements that compare data in as many ways they can. Statements and comparisons will vary. Have students use the same equation but use the numbers, 5, 10, and 3. (5 + 10)/5 = 3 Would changing the scale to a 2-interval scale work better for the 15, 5, and 20 bars? No, it would not be better because the 2 interval scale would only evenly match the 20 bar. The 15 and 5 bars would not match on a 2-interval scale. Bring in a grocery store or department store circular advertisement and have students find as many different ways to spend $9.00. Combinations will vary depending on the advertisement. Grade 4 Math Expressions Check Your Understanding Questions 14 Check Your Understanding Questions Unit 4- Angles and Polygons Grade 4 Unit Lesson 4 1 4 4 4 Activity 1 Page Number 439 2 440 3 4 442 444 1 448 2 450 1 456 2 458 3 459 Can you draw an obtuse isosceles triangle? Sample obtuse isosceles triangle: Math Connection 460 What is it called when you arrange, group, or sort objects by a set of characteristics? classify 1 464 2 466 When you build a parallelogram from triangles, do you have to use two congruent triangles (for example, 2 obtuse triangles) or can you use 2 non-congruent triangles (for example an obtuse triangle and a right triangle)? You have to use 2 congruent triangles because the triangle edges have to meet up to form the edges of a parallelogram. When you divide a parallelogram on it’s diagonal, what 2 3 4 Check Your Understanding What is the name of an angle that is smaller than a right angle? Acute angle What measurement tool is used to measure angles? protractor What is a 180° angle called? a straight angle What kinds of angles do the lines of the compass on Student Activity Book page 185 form? right angles When you rotate a half turn clockwise or counter clockwise around a clock, what kind of an angle do you make? Straight angle or 180° angle Draw this figure and ask students if the figure has rotational symmetry. Then have them write the number of degrees of the rotation. Yes; 180° True or false? A right triangle has 3 right angles? False, only one angle in a right triangle can be a right angle. Draw this triangle and ask students what the little marks mean. The little marks on the sides of the triangle tell you that the sides are congruent. Grade 4 Math Expressions Check Your Understanding Questions 15 4 4 5 figures are formed? 2 triangles Can you have more than one route between two places? There is usually more than one route between point A and point B, but people usually choose to go the shortest route. Extension 468 1 472 2 473 3 475 1 2 480 482 True or false, a polygon is a closed figure. False Draw this figure and ask students to describe it as concave or convex. concave 3 483 How would you find the perimeter of an octagon? P=8 × s 1 488 Draw a 270° clockwise rotation about Point A. How do you find the perimeter of an equilateral, an isosceles, and a scalene triangle? 3b = P; 2b + (c) = P; b+c+d=P How do you find the area of a right triangle? A= ½ × length × width When measuring triangles, if the measurement is not a whole number, what can you do? You can round the measurement to its nearest whole unit of measure. 6 A 2 489 Have students fill in the blank to make this statement true. slide Another name for a translation is called a ________. 3 4 490 7 4 Extension 491 492 Have students fill in the blank to make this statement true. reflection A ____________ is the movement of a figure across a line producing a mirror or congruent image of the figure. Have students fill in the blank to make this statement true. congruent For two figures to be ___________, they must be exactly the same size and shape. Continue this pattern and describe the rule with the word translation, rotation, or reflection. translation Grade 4 Math Expressions Check Your Understanding Questions 16 Check Your Understanding Questions Unit 5- Multi-Digit Multiplication Grade 4 Unit Lesson 5 1 5 2 Activity 1 Page Number 502 2 503 1 508 2 510 1 514 Check Your Understanding Write out the steps for finding 4 x 60 by factoring the tens. 4 x 60 = (4 x 1) x (6 x 10) = (4 x 6) x (1 x 10) = 24 x 10 = 240 Write out the steps for finding 30 x 40 by factoring the tens. 30 x 40 = (3 x 10) x (4 x 10) = (3 x 4) x (10 x 10) = 12 x 100= 1, 200 How is 30 x 40 similar to 3 x 4? 30 x 40 is 120 and 3 x 4 is 12 How is 70 x 40 similar to 7 x 4? 70 x 40 is 280 and 7 x 4 is 28 Use the Area Model to multiply 34 x 3 on your MathBoard. 102 30 + 4 3 5 3 2 516 4 x 3= 12 30 x 3 = 90 (in the first rectangle, there are 30 dots across by 3 dots down; in the second rectangle there are 4 dots across by three dots down) Use Rectangles to multiply 4 x 56. 224 50 + 6 6 × 4 = 24 50 × 4 = 200 Math Connection 518 Have students solve this word problem and explain the method they used. $336; Methods will vary. A train ticket costs $42. How much will 8 tickets cost? 5 4 1 522 2 525 Use rounding frames to estimate the product of 42 x 4. Rounding frames: 40 x 4 = 160 and 50 x 4= 200; the product is closer to 160. Estimate this product and then solve to check that your estimate is reasonable. 4 x 70 = 280; 4 x 67 = 268 so estimate was reasonable. 4 x 67 1 530 Use the Expanded Notation Method to multiply 36 x 4. 144 36 = 30 + 6 4 5 5 4 30 36 = 30 + 6 × 4 4 4 x 30 = 120 4 x 6 = 24 144 + 6 Grade 4 Math Expressions Check Your Understanding Questions 17 5 2 532 Use the Algebraic Notation Method to multiply 36 x 4. 4 36 = 4 (30 + 6) = 120 + 24 = 144 1 536 Use the Shortcut Method to multiply 36 x 4. Step 1 Step 2 2 2 36 36 × 4 × 4 4 144 2 538 Have students use any method they choose to multiply 78 x 6. 468; Check students’ methods. 1 542 Draw an area model to multiply 5 x 300. 1, 500 6 300 = 5 2 543 200 + 5 x 200 = 1, 000 100 5 x 100= 500 5 Use the Rectangle Sections Method to multiply 4 x 568. 2, 272 5 500 7 4 3 545 Math Connection 546 + 60 + 60 × 4 = 240 500 × 4 = 2,000 8 8×4= 32 Have students use any method they choose to multiply 587 x 3. 1, 761; Check students’ methods. Have students solve this problem. $35.96 Each small pizza costs $8.99. How much would it cost for a family to buy 4 pizzas? 1 551 Use the Expanded Notation Method to multiply 568 x 4. 2,272 568 = 500 + 60 + 8 4 4 500 5 + 60 + 8 568 = 500 + 60 + 8 × 4 4 4 x 500 = 2, 000 4 x 60 = 240 4x 8= 32 2, 272 8 2 552 3 555 Extension 558 Use the Shortcut Method to multiply 364 x 5. 1, 820 Step 1 Step 2 Step 3 3 2 3 2 3264 3 64 3 64 × 5 × 5 × 5 0 20 1, 820 Use rounding frames to estimate the product of 846 x 3. Rounding frames: 800 x 3 = 2,400 and 900 x 3 = 2,700; the product is closer to 2,400. Have students use any method they choose to multiply Grade 4 Math Expressions Check Your Understanding Questions 18 6, 876 x 4. 27, 504; Check students’ methods. 1 562 Read aloud this story problem and have students identify the extra information and solve the problem. The extra information is about his brother; 101 sports’ cards Brendan has 45 baseball cards and 56 football cards. His brother, Damien, has 36 baseball cards. How many sports’ cards does Brendan have altogether? 5 9 2 564 Read aloud this story problem and have students identify the missing information. The missing information is how much money she earned the rest of the month to see if she has $19.99 saved up. Melinda has saved $6.50 already. She saved some more during the month. Does she have enough to buy the doll for $19.99? 1 5 5 568 10 Identify the hidden information and solve this problem. the hidden information is the number of days in a week (7); 21 days Tamara was away for a week. Kyle was away for two weeks. How many days were both kids away for? Usually, what are the two things you must place in your answer when solving a word problem? The numerical answer and the correct label. 2 570 1 574 Use the Rectangle Model to multiply 34 x 56. 1,500 + 180 + 200 + 24 = 1, 904 56 = 50 + 6 34 30 × 6 = 30 × 50 = = 180 1,500 30 + 4 50 × 4 = 200 6 × 4 = 24 2 576 Have students use an area drawing to multiply 58 x 37. 2, 146; check students’ models. 1 582 2 585 When drawing your rectangles in a two-digit by twodigit multiplication model, what is different compared to a one-digit by one-digit multiplication rectangle model? You have 4 rectangles instead of 2 because you have to break both numbers up into tens and ones. Use the Shortcut Method to multiply 36 x 52. 1, 872 Step 1 11 1 5 12 36 ×5 2 2 Step 2 1 36 ×5 2 72 Grade 4 Math Expressions Check Your Understanding Questions 19 Step 3 3 1 36 ×5 2 72 0 Step 4 3 1 36 × 52 72 180 Step 5 3 1 36 × 52 72 +180 1, 8 7 2 5 1 590 2 591 3 593 Problem Solving Strategy 594 13 Have students solve 67 x 45 using any of the 4 methods (Rectangle Sections, Expanded Notation, Algebraic, and Shortcut) and explain their thinking. 3, 015; Check students’ methods and explanations. Use rounding to estimate the product of 34 x 56. 30 x 60 = 1, 800 Use rounding to estimate the product of 65 x 27 and then solve. Write whether your estimate was an overestimate or an underestimate. 70 x 30 = 2, 100; 1,755; overestimate Have students solve this word problem. Estimate the amount needed and then find the exact cost. $15 x 30 = $450; $389.74 The movie tickets cost $14.99 each. How much would it cost for 26 fourth graders to attend the movie? 5 14 1 598 2 600 Math Connection 602 Have students solve 76 x 23 using the shortened Expanded Notation or Shortcut method. 1,748; Check students’ methods and make sure they have reduced and dropped some of the written steps. Use rounding to estimate the product of 45 x 88. 50 x 90 = 4, 500 Use Lattice Multiplication to multiply 36 x 47. 1, 692 3 1 6 1 2 2 2 4 4 2 7 4 1 6 9 2 Grade 4 Math Expressions Check Your Understanding Questions 20 5 5 15 16 1 606 2 608 1 613 2 614 1 618 Have students use any method to solve 23 x 400. 9, 200; Check students’ methods. Why do problems that include an even number times 5 appear to have an extra 0? The even 5s products end with a 0 already. Have students use rectangles to multiply 3 x 4, 000. 12, 000; Check students’ models. Have students use any method to solve 33 x 7, 000. 231, 000; Check students’ methods. Have students place the following years on the timeline on Student Activity Book page 257. Check students’ timelines for correct placement. 1908 5 2 619 3 620 1915 1939 1955 1962 Why do you need a key on a double bar graph? The key tells you what the colored bars stand for. Have students solve this problem. 22 pencils A pencil box contains 36 pencils. Some of the pencils have an eraser. The box contains 8 more pencils with erasers than without erasers. How many pencils have erasers? 17 4 620 5 620 What does congruent mean? Congruent means having the same size and shape. Use the rectangle method to find the product of 46 x 39. 1, 200 + 180 + 360 + 54 = 1,794 46 = 40 + 6 39 40 × 30 = 30 × 6 = = 1,200 180 30 + 9 6 × 9 = 54 40 × 9 = 360 Grade 4 Math Expressions Check Your Understanding Questions 21 Check Your Understanding Questions Unit 6- The Metric Measurement System Grade 4 Unit Lesson Activity 1 Page Number 628 6 1 2 3 630 632 1 636 2 638 3 639 Problem Solving Strategy 640 1 644 2 647 3 648 1 2 652 653 3 654 1 658 2 659 3 660 6 6 6 6 2 3 How many square centimeters are in 2 sq dm? 200 sq cm Look at Student Activity Book page 268. How can you find the area of the parts of the school shown? You can measure the side length of the library (w) and measure the combined side lengths From Mrs. Lee’s room to the end of library (l) and multiply them together. What is the difference between measuring distance and area? When measuring distance, you can use the base metric unit- the meter, but when measuring area, you can use the base metric unit- the square meter. Look at problem 3 on Student Activity Book page 270. If Adita used 2 yellow tiles for every 2 red tiles, how many of each color did she use? Hint: Use a simpler problem (draw one tile) to show this. We know Adita used 9 (4-square designs) so 9 x 2 = 18. 18 red and 18 yellow equal 36 tiles. Sample drawing shown: 4 5 Check Your Understanding What are the three different markings within a meter stick? A decimeter, a centimeter, and a millimetre How many centimeters are in 2 meters? 200 cm Which is the best unit you would use to measure the distance between states? Kilometer What is the formula for finding the volume of a centimetre cube? Volume = length × width × height What is the basic metric unit for measuring volume? Cubic meter What is the basic metric unit for measuring capacity or liquid volume? Liter What is the basic metric unit for measuring mass? gram What other types of graphs could have you used to show the data on Student Activity Book page 274? Possible graphs: vertical bar graph, circle/pie graph, pictograph, line plot If you were measuring mass, would you ever use the unit pound? No, pound refers to weight, mass refers to the metric base unit gram. What are the two temperature measurement scales? degrees Fahrenheit or degrees Celsius Does 32°F and 0°C feel different? No, they are the same temperature (the temperature where water freezes). Have students name as many tools as they can and explain what those tools measure. Possible tools: scale (weight or mass); thermometer (temperature); ruler (length), cup (capacity); clock (time), protractor (angle measurement) Grade 4 Math Expressions Check Your Understanding Questions 22 Check Your Understanding Questions Unit 7- Multi-Digit Division Grade 4 Unit Lesson Activity 1 Page Number 668 Check Your Understanding Write this division example on the board and have students label which numbers are the divisor, the quotient, and the dividend. 5- divisor; 80- quotient; 400- dividend 5 7 2 674 3 675 Math Connection 676 1 80 400 What is a remainder in a division problem? Sample explanation: A remainder is the leftover quantity that is not large enough to make another whole group. True or false, a remainder can be larger than the divisor. False, a remainder must be smaller than the divisor because if it was the same as or larger than the divisor, another group could be made. Have students solve this problem. $40 Talia earned $200 in 5 days. If she earned the same amount of money each day, how much money did she earn in one day? Have students solve this problem using any division method. 53 packages 7 1 680 2 682 2 A vegetable stand sells packages containing 1 cucumber, 1 potato, 1 squash, 1 bunch of broccoli, and 1 pepper each. One week they sold a total of 265 vegetables. How many packages did they sell? How can you check a division problem. Multiply the divisor by the quotient and add the remainder. The answer should be the dividend. Have students use the partial quotients method to divide 678 by 7. 7 Math Connection 686 678 630 48 42 6 90 6 96 96 R6 1 7 690 Have students solve this problem using any division method. 40 packs A store clerk suggests selling cans of tomato sauce in packs of 4. How many packs of 4 can be made from 162 cans of tomato sauce? 3 2 693 Math Connection 694 Have students divide 2, 346 by 4 using any division method. 586 R2 What is 70 ÷ 10 and 700 ÷ 10? 7 and 70 Grade 4 Math Expressions Check Your Understanding Questions 23 7 4 1 698 2 699 700 Math Connection 7 7 1 704 2 707 Problem Solving Strategy 708 1 712 2 714 3 718 5 6 Why did you know that 7 was not the right number in the Puzzled Penguin’s quotient on Student Activity Book page 289? The difference (6) was larger than the divisor (4) so you knew you could make another group. What is 6, 570 ÷ 9? 730 When do you think you should choose mental math over using a calculator to help solve a problem? You may choose mental math if the problem can be done using a simpler problem or if it follows a pattern. Solve using any method. Then check your answer by rounding and estimating. 1, 892 R2; Check student’s methods; estimates: 6, 000 ÷ 3 = 2,000 5, 678 ÷ 3 Solve using any method. Then check your answer by rounding and estimating. 765; Check student’s methods; estimates: 4, 000 ÷ 5 = 800 3, 825 ÷ 5 When you estimate, does your answer have to be correct? No, estimating is just trying to make a close guess of what you think the answer or measurement will be. Some answers can be estimates while others need to be exact. There are many types of remainders. Explain these types of remainders. 1. left-over remainder: the remainder cannot be used in the context of the problem and is ignored. 2. another whole remainder: the remainder causes the answer to be rounded up. 3. fractional part remainder: the remainder can be fractured and shared (i.e. things to eat) and we write the answer as a whole number and a fraction. 4. a decimal remainder: the remainder can be shared (i.e. money) and we write the answer as a decimal. 5. remainder only: the remainder is the only part needed to answer the questions (i.e. the extra person who cannot participate). Divide and write the remainder as a fraction. 4 2/4 or 4 1/2 18 ÷ 4 Have students solve this problem. 8 R3, so 9 boxes. The remainder makes it necessary to buy 9 boxes instead of 8. Davio wants to buy some new baseball card boxes. He has 67 cards. Each box holds 8 cards. How many baseball card boxes does Davio need to buy? 1 7 722 7 2 724 Have students write solution and situation equations for this problem. Solution: b = 3 + 2; situation: b – 3 = 2 Marla had some oranges. She gave 3 oranges to her friends. Now she has 2 oranges. How many oranges did have in the beginning? Have students solve this problem and explain their Grade 4 Math Expressions Check Your Understanding Questions 24 methods. 360 – 10 = 350; 350 – 3 = 347; 350 + 347 = 697; 697 balls A team from each school had 360 foam balls and a bucket. The Springfield team dunked 3 fewer balls than the Oak Hill team. The Oak Hill team dunked all but 10 of their balls. How many balls did the two teams dunk in all? 7 7 7 8 1 728 2 730 3 733 Math Connection 734 1 739 How do you find the mean of a set of data? To find the mean or the average, you add the values in the set of data and divide that sum by the number of values. What is the mode, when describing data? The mode is the value that appears most frequently in a set of data. How do you find the range of a set of a data? The range is the difference between the greatest number or maximum value and the least number or minimum value in a set of data (greatest number/max value – least number/min value = range). If a number is divisible by another number, is there a remainder? No there is no remainder. Divide the following to see the pattern. 7, 7, 7, 7 7÷1 70 ÷ 10 700 ÷ 100 7,000 ÷ 1,000 9 Extension 740 What is 3 x 12? 36 1 744 2 745 3 746 What does a digital clock usually tell us that an analog clock does not show? Whether the time is a.m. or p.m. How many hours have passed from 9:30 am to 11:00 am? 1 ½ hours Calculate this elapsed time. 10 9:50 a.m – 8:33 a.m. Math Connection 748 1 752 1 h and 17 min How many years is a century? 100 years Use the Identity Property to simplify this expression. 15f 13f + f + f = ______ 7 7 11 12 2 755 What are the Order of Operations? When solving equations that include parentheses, first perform the operations inside the parentheses. Second, multiply and divide from left to right. Then, add or subtract from left to right. 1 760 Have students identify which one of these examples is NOT an expression. a + 3 = 6 16 b a+3=6 6c f÷5 6–c+b Grade 4 Math Expressions Check Your Understanding Questions 25 7 13 2 762 True or false, an inequality is a statement that two expressions are equal. False, an inequality shows that two expressions are not equal. 1 766 Solve this single step equation. y = 4 2 768 4y = 16 Solve this two-step equation. y = 4 4y + 3 = 19 7 25 1 774 Have students draw a triangle with a line of symmetry on their MathBoard. Sample drawing: 2 3 775 776 4 776 5 776 Which would you prefer? 3 x $1 or $1, 000 x 0? 3 x $1 If you support an argument, what does that mean? It means you agree with an argument and you use evidence to back up your argument. What are some different rectangles (with whole number sides) that have an area of 24 square centimeters? What is the perimeter of each? 1 x 24 or 24 x 1 rectangle (P =50 centimeters); 2 x 12 or 12 x 2 rectangle (P =28 centimeters); 3 x 8 or 8 x 3 rectangle (P =22 centimeters); 4 x 6 or 6 x 4 rectangle (P = 20 centimeters) Using your estimate from the 200th person in line, how long can you assume that the 400th person in line must wait? 3 hours and 20 min Grade 4 Math Expressions Check Your Understanding Questions 26 Check Your Understanding Questions Unit 8- Patterns, Functions, and Graphs Grade 4 Unit 8 8 Lesson Activity 1 Page Number 785 Check Your Understanding Draw the next figure in the pattern. 2 786 Draw this growing pattern on the board and ask students to decide what comes next. Extension 788 What shapes are formed by cutting this figure into parts? Sample shapes: 1 792 Describe the pattern and identify the next term in it. Subtract 4; 72 2 793 1 2 96, 92, 88, 84, 80, 76, … Have students fill in the blanks to make the statement true. Addition; multiplication; subtraction; division If the terms in a pattern are becoming larger, the pattern is likely to involve _________ and _________. If the terms in a pattern are becoming smaller, the pattern is likely to involve _______ and _________. 8 8 1 798 2 799 3 801 1 807 3 What does inverse operations mean? The opposite operation What describes the mathematical relationship shared by two sets of numbers? A rule When you are trying to find the rule of a function table, can you only test the first set of input and output values. No, you must test the rule to each pair of numbers. Have students fill in the blanks to make the statement true. x; y The first coordinate represents distance along the _______ axis. The second coordinate represents distance along the _______ axis. 4 2 808 3 810 Name two points on Student Activity Book page 332 that form a straight angle. Sample angle: <CW Have students draw another square or rectangle on the coordinate plane on Student Activity Book page 333. Then have pairs find the length of line segments. Grade 4 Math Expressions Check Your Understanding Questions 27 8 8 5 Extension 812 1 816 2 818 3 820 Extension 822 1 826 2 830 Answers will vary. Look at problem 1 on Student Activity Book page 334. Translate the top square 2 units to the right and write the vertices of the translated square. (4, 7), (7, 7), (4, 10), (7, 10) What type of function is shown on Student Activity Book page 335? Linear function Sometimes the letters in a function table may not be x and y. In this example, the variables c and p stand for which axes? C is the x-axis and p is the y-axis. How can you predict other points on a linear graph? Use the pattern they see in the other ordered pairs. Have students draw two other points (C and D) on the grid on Student Activity Book page 338. Then have them choose and describe a path between both points. Paths will vary. Check students’ descriptions. What types of data does a line graph show? Change over time data Have students fill in the blanks to make the statement true. scale 6 Since it’s difficult to graph points that are between grid lines, it’s best if all or most data points fit exactly on the ______________. Grade 4 Math Expressions Check Your Understanding Questions 28 Check Your Understanding Questions Unit 9- Fractions Grade 4 Unit Lesson 9 1 9 2 Activity 1 2 Page Number 838 840 Check Your Understanding What does the term fraction mean? Part of a whole Write ¾ on the board and have students write the word name for the fraction. Then have them sketch a representation of the fraction. Three fourths; sample drawing: 3 842 Have students give an example of unit fractions. Possible unit fractions: ½, 1/3, ¼, 1/5, 1/6, (any fraction with a 1 as the numerator), etc. 1 2 848 850 What does 5/5 equal? 1 Find the unknown partner in this equation.4/6 2/6 + d = 6/6 9 1 854 2 856 3 858 3 Have students complete this fraction-partner equation. 4 sevenths; 4/7; 7 sevenths equals 3 sevenths plus 4 sevenths. 3/7 + [ ] = 7/7 Write these two fractions on the board and have students compare them with the greater than, less than, or equal to symbols. 1/5 < 1/3 Write these two fractions on the board and have students compare them with the greater than, less than, or equal to symbols. 4/5 1 864 > 3/10 Draw this on the board and ask students to decide if the statement is true or false. False, since the whole pies are different sizes, the three pieces in each pie are not equal in size. Hettie says that ¾ of pie A is the same as ¾ of pie B. 9 9 4 5 2 865 1 870 A B Have students share their word problems they wrote and have the class solve them and explain their thinking. Problems and answers will vary. What is the diameter of the circle? The length of a line Grade 4 Math Expressions Check Your Understanding Questions 29 9 9 segment that goes from one side of the circle to the other and passes through the center. What is the radius of the circle? A line segment that connects the center of a circle to any point on that circle. Also the length of that line segment. What is the circumference of the circle? The distance around the outside of a circle. 2 872 Extension 874 1 878 Have students divide a circle into 6 equal pieces. Sample circle: 2 879 Extension 882 True or false, all pieces of a circle graph have to be equal. False, each piece of the graph shows a part of the whole and can be different sizes. What percentage do all the pieces of a circle graph add up to? 100% 1 886 2 888 1 892 6 7 Use fraction bars to add 3/10 and 4/10. 3/10 + 4/10 = 7/10 Use fraction strips to subtract 1/5 from 4/5. 4/5 – 1/5 = 3/5 Add 4/6 and 2/6 with Class Fraction Cards. 6/6 1/6 9 8 2 893 3 896 1/6 1/6 1/6 9 + 1 903 Add these improper fractions. 12/6 2 905 8/6 + 4/6 Subtract these mixed numbers. 1 1/5 3 906 2 4/5 – 1 3/5 Write these mixed numbers on the board and have students use a greater than, less than, or an equal to symbol to compare the mixed numbers. 2 1/2 9 9 10 1 910 915 2 918 11 > 2 1/4 Have students identify the greater fraction. ¾ ¾ 1 1/6 What is a mixed number? A mixed number is a whole number “mixed” with a fraction. Draw this on the board and have students write the mixed number and improper fraction the drawing shows. 1 + 1 +1/3 = 2 1/3; 3/3 + 3/3 + 1/3 =7/3 + 9 1/6 + 4/8 Use division to find an equivalent fraction for 4/6. Sample equivalent fraction: 2/3; 4/6 = 4 ÷ 2/ 6 ÷ 2 = 2/3 Complete this fraction equation. Grade 4 Math Expressions Check Your Understanding Questions 30 9 9 3 920 1 925 2 929 3 930 1 934 2 935 3 936 12 13 1 × 6 =6 3 × 6 18 Simplify the fraction 9/12. 3/4; 9/12 = 9 ÷ 3/ 12 ÷ 3 = ¾ Use fraction bars to add ½ and 2/8. ½ = 4/8, 4/8 + 2/8 = 6/8 Use the Multiplication table to find an equivalent fraction for 2/3. Sample equivalent fraction: 4/6; 2/3 = 2 x 2/ 3 x 2 = 4/6 Find a common denominator to add these fractions. 6/8 + 1/8 = 7/8 3/4 + 1/8 Find equivalent fractions for 1/6 and 1/3 and then subtract. 2/12 and 4/12; 4/12 – 2/12 = 2/12 Write these fractions on the board and have students use a greater than, less than, or an equal to symbol to compare the fractions. Hint: find equivalent fractions before you compare. 1/5 < 1/3 Order the fractions from least to greatest. 1/8, 2/4, ¾ ¾, 1/8, 2/4 9 1 940 2 941 14 Math Connection 942 For this pair of fractions, find an equivalent fraction, then compare the fractions, add them, and then subtract them. 1/8 and 4/8; 1/8 < 2/4; 5/8; 3/8 1/8, 2/4 Have students solve this word problem. Sonja cannot make this statement from picking only 5 marbles. She can only make a prediction. Sonja has 50 red and yellow marbles. She pulls out 5 marbles, returning each to the bag after she pulls it out. If she pulls out 2 red marbles and 3 yellow marbles, can she say that there are 20 red marbles and 30 yellow marbles? Subtract these mixed numbers. 5 6/8 – 3 4/8 = 2 2/8 or 2 ¼. 5 6/8 – 3 2/4 9 15 1 947 2 949 If you repeat the same experiment two times in a row, will you get the exact same results? Not likely, as probability tells us what’s expected to happen, not exactly what will happen. Draw the following on the board and tell students that figures represent boxes of white and black marbles. box A; 5/9; 6/9 A B Suppose you take one marble from box A without looking and one marble from box B without looking. Grade 4 Math Expressions Check Your Understanding Questions 31 From which box are you more likely to get a white marble? What is the probability of choosing a black marble from box B? 3 9 9 9 16 950 What is the probability of choosing a white marble from box A? What is the probability the arrow will land in the space marked B? 3/4 B B B 4 951 What does the word “outcome” mean? Outcome means the result of an experiment or what comes out. What makes a game fair or unfair? A game is fair if each person has an equal chance of winning. A game is unfair if one person has more of a chance of winning. Math Connection 954 1 959 2 962 1 966 Multiply 3 x ¼ and draw a shape to show the solution. ¾ 2 967 Solve 7 x 2/3 =. 14/3 1 2 972 973 Solve 1/8 x 16 =. 2 Solve 3/8 x 16 =. 6 1 978 Have students make comparison statements about these comparison bars. Ariel has 4 times as many comic books as Pasqual. Pasqual has ¼ as many comic books as Ariel. Which graph would you use to display how often things occur? line plots What is the mode, when describing data? The mode is the value that appears most frequently in a set of data. 17 18 Ariel Pasqual 9 A 19 2 979 3 980 6 6 6 6 6 Have students share their comparison bars from problem 6 on Student Activity Book page 398. Then have the class state each comparison both ways. Comparisons will vary depending on the two rides being compared. Have students solve this word problem. 15 hours Stephanie practiced keyboard for 5 hours this week. She spent 1/3 as much time practicing as Raul did. How many hours did Raul practice this week? Grade 4 Math Expressions Check Your Understanding Questions 32 9 9 20 1 986 2 987 1 992 2 994 21 3 994 1 998 2 999 3 1000 Label a new point (I at 1 3/8 or 11/8) on a number line on Student Activity Book page 401. Point I should be located on the 4th number line down on the tick mark right before 1 4/8. Label a new point (I at 9/10) on number line 10 on Student Activity Book page 402. Point I should be located on the number line on the tick mark right before 1. Write these fractions on the board and have students use a greater than, less than, or an equal to symbol to compare the fractions. 2/8 < 3/6 Have students solve this word problem. 24 books Dahlia has 36 books. She’s read 2/6 of them already. How many books hasn’t she read? Label a new point (K at 5 1/8) on number line 37 on Student Activity Book page 404. Point K should be located on the number line on the tick mark right after the 5. Use a hexagon pattern block to make a tessellation pattern. Check student’s tessellations. What are the two types of data students can collect? numerical or categorical data Can you make a generalization from your doubling proof, that tripling the data would yield the same results to the mode and the range? Give an example to back up your generalization. Generalization: If the values in a group of data are tripled, then the mode and the numbers that show the range are also tripled. Original data {2, 5, 5, 7, 8} Mode: 5; Range: 6 4 9 1000 22 5 1000 Tripled data {6, 15, 15, 21, 24} Mode: 15; Range: 18 Describe a sample type of data each graph may show. Answers may vary. Sample data shown. Bar graph: favourite hobbies Pictograph: Books checked out at the library Line graph: temperature change Line plot: number of siblings Circle graph: types of birds Use the whole numbers and fractions you have already written on the board and have students use these clues to find out which fraction you are describing now. 4/7 2/5 1/8 4/7 7/10 5 ¾ The numerator is an even number and the denominator is an odd number. The sum of the denominator and the numerator is 11. Which could be Sam’s number? Grade 4 Math Expressions Check Your Understanding Questions 33 Check Your Understanding Questions Unit 10- Three-Dimensional Figures Grade 4 Unit Lesson 10 1 10 10 2 Activity 1 Page Number 1008 2 3 1010 1012 1 1016 2 1018 3 1019 Math Connection 1020 1 1024 2 1025 3 1026 3 Check Your Understanding How is a sphere different from a circle? A sphere is 3-D and a circle is 2-D. How many squares make up a cube net? 6 squares How do you know there are hidden cubes even if you can’t see them? The hidden cubes are holding up the cubes that you can see. Why do you think all your prism nets are made from a different amount of separate rectangles? Depending on what type of prism you are folding determines the number of rectangular faces you have. For example, a pentagonal prism has 5 rectangular faces plus the two pentagon faces. How are a cylinder and a cone alike and different? Both figures have circular bases, but a cone has only one base and a point on top. How do you find the surface area of a solid figure? It’s the total area of all of its faces. What is a net? A 2-D pattern that can be folded into a 3-D figure. What is a face, an edge, and a vertex? Face: a plane (flat figure); edge: side of a face; vertex: a corner where edges meet. True or false, a pyramid has a pair of congruent bases. False Using the pattern on Student Activity Book page 418, find the number of sides on base, number of faces, number of edges, and number of vertices on an octagonal prism. 8, 10, 24, 16 Grade 4 Math Expressions Check Your Understanding Questions 34 Check Your Understanding Questions Unit 11- Decimal Numbers Grade 4 Unit Lesson 11 1 11 2 11 3 Activity 1 Page Number 1036 2 1044 1 1054 Write this number in decimal form. 0.22 22 hundredths 1 1064 Write the word name of the number. Six tenths 0.6 1 11 Check Your Understanding What unit fraction can you write to represent how many dimes are in a dollar? 10/100 or 1/10 Write a fraction and a decimal to represent 80 cents out of a dollar. 80/100 or 8/10; 0.80 or 0.8 1068 4 2 1074 Have students fill in the blanks to make the statements true. multiply; divide As you move to the left of the place value chart (towards the tens and hundreds), each place gets bigger and you __________ by 10. As you move to the right of the place value chart (towards the tenths and hundredths), each place gets smaller and you __________ by 10. Write the equivalent number of hundredths and thousandths. 0.70; 0.700 0.7 11 5 1 1082 2 1084 Where can a zero be inserted in a decimal number without changing the value? Zeros can be inserted to the right of the number without changing the value. Write these decimals on the board and have students use greater than and less than symbols to compare the decimals. 0.45 < 0.54 11 11 1 1088 2 1090 1 1094 2 1098 6 7 Draw a square at 0.67 on problem 5’s number line on Student Activity Book page 439. Then round this hundredth to the nearest tenth. Check children’s square location; 0.7 What is $3.50 rounded to the nearest dollar? $4.00 Continue to call out more whole numbers and have students plot and label the numbers at those positions. Numbers and positions will vary. Draw a 0 – 3 number line on the board and have students plot 1.5 and 2 ¾ on that number line. Check number lines. Relative locations of points shown: 0 11 8 1 1102 1 1.5 2 2¾ 3 Have students solve this problem. $0.63 Dario has 2 quarters, a dime and 3 pennies. What Grade 4 Math Expressions Check Your Understanding Questions 35 decimal part of a dollar does he have? 1 1106 Display a 10 x 10 grid on the overhead and represent this addition. Then solve. 0.5 + 0.3 = 0.8 0.5 + 0.3 11 9 2 1108 Solve this problem. 1110 3.157 .67 3.827 Align decimal numbers to add. $35.78 + 25¢ + 3 + 1 1114 35.78 .25 $ 36.03 Display a 10 x 10 grid on the overhead and represent this subtraction. Then solve. 0.5 – 0.3 = 0.2 0.5 – 0.3 11 10 2 1116 3 1118 Find the difference between these numbers. 0.46 0.76 and 0.3 Have students solve this problem. $2.66 Mrs. Grande has earned $2.34 so far. How much more money will she need to earn to have $5.00? 11 1 1123 2 1125 Math Connection 1126 11 Write the amount for 3 quarters, 1 dollar, and 5 nickels. $2.00 Have the students solve this problem by writing the names and numbers of coins and bills you could use to make change. Coins for change may vary; 2 quarters, 1 dime, 4 pennies The cost of an item is 36¢. You give the clerk a dollar. Write the next term in the sequence. 51.4 52.4, 52.2, 52.0, 51.8, 51.6 _________ Grade 4 Math Expressions Check Your Understanding Questions 36 11 1 1130 2 1132 Math Connection 1134 12 Estimate the answer by rounding each number to the nearest tenth. 0.8 + 0.4 = 1.2 0.76 + 0.35 Select a rounding unit that allows you to estimate the answer mentally. Write your rounded problem and your estimated answer. 0.8 + 0.3 = 1.1 0.765 + 0.25 Estimate the answer by rounding these metric measurements to the nearest tenth. 6.5 + 2.3 = 8.8 6.45 grams + 2.345 grams 11 13 1 1139 Math Connection 1140 Order these decimals from greatest to smallest. 15.6, 12.2, 4.5, 0.675 12.2 15.6 0.675 4.5 Multiply. × 11 11 14 1 1144 2 1148 1 1152 2 1153 3 1154 4 1154 $ 3.15 6 $ 18.90 Is 2/5 closer to 0, ½, or 1? 2/5 is closer to ½ as 2/4 is equal to ¼ and 2/5 is very close to 2.4. Use the table on Student Activity Book page 460 to estimate these measurements. 1 5/8 + 3/4 = 1 ½ + 1 = 2 ½ cups. wheat flour + white flower Have students show their final building design from Student Activity Book page 461 and have the class describe the building including the names of the geometric solids used. Buildings and descriptions vary. Color this drawing so that the colors of sections that are next to each other are not the same color. How many colors did you use? 3, sample color pattern shown: 15 5 1154 Write 7 + 4 < 6 + ______ on the board. Have students use reasoning and the process of elimination to decide which number (3, 4, 5, 7) makes the statement true. 7 Use logical reasoning and the clues to find the heights of these dogs. Then make a bar graph to graph your height data. Darien: 45 in.; Clara: 15 in.; Toby: 20 in.; Brenda: 30 in.; Check bar graphs. Darien’s height is 3 times the height of Clara. Clara’s height is 5 inches shorter than Toby. Toby’s height is 20 inches. Brenda’s height is 2 /3 the height of Darien. Do 0.05 and 0.5 have the same value? No, 0.05 is 5 hundredths and 0.5 is 5 tenths. Grade 4 Math Expressions Check Your Understanding Questions 37 Check Your Understanding Questions Unit 12- The U.S. Customary System Grade 4 Unit 12 12 Lesson Activity 1 Page Number 1162 2 3 1164 1165 Problem Solving Strategy 1166 1 2 1170 1171 3 1172 1 2 Check Your Understanding Which is the best unit you would use to measure the distance between states? mile How many inches are in a foot? 12 inches People used to say an inch was the size of a man’s thumb. How could knowing this help you with estimation? You could use your thumb to estimate length if you did not have a ruler. Have student volunteers explain how they used the Word Backward strategy to solve problems 2-4. Observe student’s methods. What is the area formula for a rectangle? A = b x h What is the formula for finding the volume of a rectangular prism? Volume = length × width × height Have students complete the statement to make it correct. square; cubic The ________ unit is always included in an area measurement. ________ units are always included in volume measurements. 12 12 12 3 1 2 3 1176 1177 1178 Which weighs less, an ounce or a pound? An ounce How many pounds equal a ton? 2,000 pounds What tool can you use to measure weight? Balance scale 1 2 3 1182 1183 1184 Which holds more, a pint or a gallon? A gallon How many quarts are in 3 gallons? 12 quarts Have students name as many tools as they can and explain what those tools measure. Possible tools: scale (weight or mass); thermometer (temperature); ruler (length), cup (capacity); clock (time), protractor (angle measurement) 1 2 1188 1189 3 Extension 1190 1192 What fraction of a year is three months? 3/12 or ¼ What Fahrenheit temperature does water freeze and boil? Freeze- 32°F and Boil- 212°F What are numbers below zero called? negative numbers True or false, integers are only positive numbers. False, integers are negative whole numbers too. 4 5 Grade 4 Math Expressions Check Your Understanding Questions 38 Unit Lesson Extension 1 Extension Extension 2 3 Activity 1 Page Number 1200 1 1208 2 1210 1 1216 2 1219 Check Your Understanding Write the percent for this fraction. 80% 80/100 Which word can you use to write the unit rate without the 1? per Change the unit rate to $5 per gallon for problem 13 on Student Activity Book page 1211 and adjust the cost. $5, $10, $15, $20, $25, $30, $35 When you read a ratio, when you see the colon (:) what word do you say for the colon? to If you added 90 to column c, what would the ratio of customers to prizes (p) be? 90:27