Grade 4 Unit 3: Multiplication and Division of up to a 4 ... · PDF fileGrade 4 Unit 3: Multiplication and Division of up to a 4-Digit Number Using Place Value (9 Weeks) Stage 1 –

Grade 4 Unit 3: Multiplication and Division of up to a 4-Digit Number Using Place Value

(9 Weeks)

Stage 1 – Desired Results

Established Goals Unit Description Compound measurement units will provide students the concrete foundation behind the distributive property in the multiplication algorithm: 4 × (1 m 2 cm) can be made physical using ribbon where it is easy to see the 4 copies of 1 m and the 4 copies of 2 cm. Likewise, 4 × (1 ten 2 ones) = 4 tens 8 ones. The Mathematical Practices should be evident throughout instruction and connected to the content addressed in this unit. Students should engage in mathematical tasks that provide an opportunity to connect content and practices. Common Core Learning Standards 4.OA.1 Interpret a multiplication equation as a comparison, e.g. interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparison as multiplication equations. 4.OA.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. 4.OA.3 Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. 4.OA.4 Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite. 4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. 4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. 4.MD.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula and the length, by viewing the area formula as a multiplication equation with an unknown factor. Common Core Standards of Mathematical Practice

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

ESL Language Standards Standard 1: Students will listen, speak, read, and write in English for information and understanding 1.4 Compare, contrast, and categorize to gain a deeper understanding of information and objects. 1.5 Formulate, ask, and respond to questions to obtain and provide information and meaning. 1.7 Present information clearly in oral and graphic forms. 1.14 Consult print and non-print resources (e.g., audio/visual media, family) in the native language when needed. Standard 3: Students will listen, speak, read, and write in English for critical analysis and evaluation. 3.4 Evaluate student’s own and others’ work, individually and collaboratively, on the basis of a variety of criteria. 3.9 Apply learning strategies to examine, interpret, and evaluate a variety of materials.

Standard 4: Students will listen, speak, read, and write in English for classroom and social interaction.

4.1 Use a variety of oral, print, and electronic forms for social communication and for writing to or for self, applying the conventions of social writing. (L, S, R, W) 4. 3.Request and provide information and assistance, orally or in writing, for personal, social, and academic purposes. (L, S, R, W) 4. 4. Listen attentively and take turns speaking when engaged in pair, group, or full-class discussions on personal, social,

and academic topics. (L, S)

4. 5.Explain actions, choices, and decisions in social and academic situations. (S, W)

4. 7. Follow oral and written directions to participate in classroom and social activities. (L, R)

4. 9. Use appropriate vocabulary, expressions, language, routine, and interaction styles for various audiences and formal and informal social and school situation. (L, S, R, W) Bridge Guidance • Solving for the unknown, using a symbol, in multiplication and division word problems with equations and in perimeter/ area problems with side lengths. • Developing the idea of area by recognizing it as an attribute of plane figures and partitioning a rectangle into rows and columns of same-size squares. • Calculating area of polygons using multiplication, distributive property, and decomposition into rectangle as strategies to solve word problems. • Recognizing and generating simple equivalent fractions.

Big Ideas 1. The same number sentence can be associated with

different concrete or real world situations AND different number sentences can be associated with the same concrete or real world situations

2. Some attributes of objects are measurable and can

be quantified using unit amounts.

3. For a given set of number there relationships that are always true, and these are the rules that govern arithmetic and algebra.

Essential Questions 1. What is the difference between a multiplicative and

additive comparison?

1. How can we solve real world problems involving joining,

separating, part-part-whole, or comparison using addition

or subtraction?

1. How can we solve real world problems involving equal

groups, separating equal groups, comparisons, or

combinations using multiplication and division?

1. How does the idea of a remainder change based on the

context?

2. How can we apply concepts of area and perimeter to

solve real-world problems?

3. How can we use place value understanding, properties

of operations, and inverse relationships as strategies to

solve problems?

Content (Students will know….) A. A multiplication equation can be thought of as a comparison. For example 35=5X7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. (4. OA.1) B. Multiplicative comparisons (also versus an additive comparison) (4.OA.2) C. A whole number is a multiple of each of its factors (4.OA.4) D. Prime and composite numbers from 1-100 (4.OA.4) E. Multi-step word problems using any of the four operations represented by equations with the letter standing for the unknown quantity (4.0A.3)

Skills (Students will be able to…) A1. Write and identify equations and statements for multiplicative comparisons. A2. Identify and verbalize which quantity is being multiplied and which number tells how many times. B1. Translate comparative situations into equations with an unknown and solve B2. Multiply or divide to solve word problems using drawings and equations B3. Differentiate between an additive and a multiplicative comparison C1. Find factors and factor pairs of numbers 1 – 100 D1. Identify and explain what prime and composite number are D2. Identify the factor of a number as prime or composite D3. Investigate, using arrays, why a number is either prime or composite based on the number of rectangles that can be produced D4. Use tools such as number lines, hundreds charts, arrays, or cubes to model relationship between factors and multiples D5. Identify factor pairs E1. Solve multi-step problems using any of the four operations E2. Solve multi-step word problems in which the remainder must be interpreted

F. Estimation/mental math computation strategies (4.OA.3) G. Multi digit multiplication (4.NBT.5) H. Multi digit division (4.NBT.6) I. Area and perimeter (4MD.3)

F1. Represent these problems using an equation with a letter for the unknown quantity F2. Use estimation strategies such as using compatible numbers or rounding to solve multi step story problems G1. Multiply up to four-digit by a one-digit whole number G2. Multiply two-digit numbers by two digit numbers G3. Multiply by using strategies based on place value and properties of operations. G3. Use tools such as arrays, and/or area models to illustrate and explain calculations H1. Compute whole number quotients with up to 4 digit dividends and one digit divisors H2. Use strategies based on place value, properties of operations, and/or the relationship b/w multiplication and division H3. Illustrate and explain calculations using equations, rectangular arrays, and/or area models I1. Identify missing length or width when the area is given I2. Calculate the perimeter of a rectangle

Terms/ Vocabulary Measurement, comparative, multiplicative comparisons, additive comparison, multiplicative equations, whole number, multiples, factors, factor pair, prime, composite, units, variables, expressions, equations, arrays, product, digit, area, calculation, dividends, divisors, place value, dimensions

Stage 2 – Assessment Evidence

Initial Task: Food Drive Final Performance Task: Recycling Water Bottles

Other Evidence Teacher observation, conferencing, teacher designed assessment pieces, student work, exit slips, journal entries

Stage 3 – Learning Plan

Everyday Mathematics/Impact Mathematics Lessons – The following lessons may support some of the CCLS & essential questions outlined in this unit map:

4OA1: 2-9, 3-3, 3-5, 3-11, 5-1 4OA2: 5-1, 5-8, 6-1, 8-8 4OA3: 2-7, 2-9, 3-8, 5-3, 5-5, 5-8, 5-11, 6-1, 6-3, 6-4, 6-8, 6-10, 8-8, 9-6, 11-7, 12-2, 12-3 4OA4: 3-2, 3-3, 3-11, 6-2, 6-4, 7-7, 7-12a, 12-3 4NBT5: 5-1, 5-4, 5-5, 5-7, 9-8, A21-A25 4NBT6: 3-5, 6-1, 6-2, 6-3, 6-4, 6-6, 6-10, 9-9 4MD3: 8-3, 8-5, 8-6, 9-2 Additional Resources: 4.OA.1

This virtual manipulative tool allows students to create arrays displaying different groupings (e.g. 3 groups of 6 and 6 groups of 3) http://nlvm.usu.edu/en/nav/topic_t_1.html/

This resource includes a short lesson and a game for students to practice building arrays, writing equations, and solving for a product http://mathstory.com/mathlessons/arrayrace.htm/

Locate the activity “Groups of Dogs” at the following site for students to look at arrays using objects

http://www.internet4classrooms.com/grade_level_help/solve_problems_math_fourth_4th_grade.htm/ 4OA.2

Look at multiplication and division word problems to use with students of multiplicative comparison http://www.helpingwithmath.com/

Challenging examples of word problems using multiplicative comparison http://www.mathplayground.com/wordproblems.html/scroll down to the correct domain and standard to find the listing of online problems displaying multiplicative comparison

4OA3

Multi step multiplication and division word problems: http://www.k-5mathteachingresources.com/support-files/4oa3multistepwordproblems.pdf

Interpreting Remainders: http://www.k-5mathteachingresources.com/support-files/interpretingremainders4.oa3.pdf

4OA4

Students distinguish between numbers with several factors and those with only a few factors.http://illuminations.nctm.org/LessonDetail.aspx?ID=L620/pdf

This link offers a worksheet that assesses students’ knowledge after playing the factor game.http://illuminations.nctm.org/lessons/FactorGame/FactorGame-AS-Problems.pdf/

This site program lists a number and the student identifies it as prime or composite http://www.aaamath.com/fra63ax2.htm/

Watch a video that defines the terms from the standard, including prime and composite http://www.aaamath.com/fra63ax2.htm/

4NBT5

Games and activities to reinforce finding the remainders, multiplication facts, writing number stories http://www.k-5mathteachingresources.com/4th-grade-number-activities.html

http://www.khanacademy.org/commoncore

4NBT6

Games and activities to reinforce finding the remainders, multiplication facts, writing number stories http://www.k-5mathteachingresources.com/4th-grade-number-activities.html http://www.khanacademy.org/commoncore

4MD3

http://www.khanacademy.org/commoncore

http://www.k-5mathteachingresources.com/support-files/fencingagarden.pdf

http://www.k-5mathteachingresources.com/support-files/designingazooenclosure.pdf

Grade 4 Unit 3 Initial Assessment: Food Drive

1. P.S. 122 is having a food drive for the month of December to donate to a local shelter. This week,

Jack and Paul donated contributed canned food for the class collection.

a) Jack contributed 9 more cans than his friend Paul. Paul contributed 8 cans. How many cans of food did Jack contribute? Solve using an equation:

b) The following week, Paul contributed 7 cans of food. Jack contributed 5 times as many cans of food as Paul. How many cans of food did Jack contribute? Solve using an equation:

c) Use numbers and words to explain how questions (a) and (b) are similar and different

2. Luis is helping Ms. Brown arrange the cans of food they collected. There are 50 large cans and 33 small cans.

a) Show 2 different ways that Luis can arrange all 50 large cans into equal size rows and columns.

Write a multiplication expression for each diagram.

Expression:_____________________

Expression:_____________________

b) How many different ways can Luis arrange all 33 small cans into equal size rows and columns? Write a multiplication expression that represents your answer.

c) There are a total 83 cans of food. How can you tell if the number is prime or composite?

Explain your mathematical thinking.

3. Class 2A collected 25 cans of food. The students will be arranging them in shoeboxes. Each shoebox fits 9 cans of food. How many shoeboxes does the class need?

a) Draw a diagram to help solve the problem

b) Write an equation using a symbol for the unknown variable.

c) How do you know your answer is correct? Explain

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

4. The school is thinking about using larger boxes to arrange the canned food. The area of the bottom of the box is 27 square inches.

a) What are the dimensions (length and width) of the bottom of the box? Use a diagram and show your work

b) What is the perimeter of the bottom of the box?

5. The table below shows the total number of cans collected in each grade:

Grade Total Cans Collected

Grade 1 152

Grade 2 71

Grade 3 275

a) What is the total number of cans collected? Show your work.

b) Grade 5 collected three times as many cans of food than Grade 3. How many cans of food did Grade 3 collect?

Show your work. Explain your mathematical thinking

Write an equation: ____________________________

c) Each student in Class 4A collected 13 cans. There are 24 students in this class. How many cans did they collect? Show your work.

Multiplication Equation: _________________________

Grade 4 Unit 3

Initial Assessment Scoring Guide

Food Drive Scoring Guide Points Section

Points

1. 4.OA.1, 4.OA.2

a) Student writes a correct equation and solves correctly

9 + 8 = 17 or 8 + 9 = 17

b) Student writes a correct equation and solves correctly

7 x 5 = 35 or 5 x 7 = 35

c) Similar: both are comparisons but “a” is an addition (additive)

comparison and “b” is a multiplication (multiplicative) comparison

1

1

1

3

2. 4.OA.4

a) Student draws two diagrams showing equal size rows and

columns and writes the corresponding multiplication expressions

such as 2 x 25 or 5 x 10 or 1x 50

b) Student correctly answers that there is only one way to arrange

and writes the correct corresponding multiplication expression

1 x 33 or 33 x 1

c) Student correctly answers that because there is only one way to

arrange the cans into equal rows and columns or because there

are only 2 factor (1 factor pair) of 1 and itself – I know it is a prime

number

2

1

1

4

3. 4OA3, 4.NBT.6

a) Student must draw a diagram that represents the 9 cans into one

shoebox and a total of 25 cans for a total of 3 shoeboxes

b) Student correctly writes the equation 25 ÷ 9 = x or 25/9 = x or 9x =

25

c) Student can explain using any variety of reasons – can reference

the diagram, can explain that there are 6 leftover so they will need

a third box, etc

1

1

1

3

4. 4MD3, 4.OA.4

a) Student must draw a diagram that shows the dimensions of either

rectangle with an area of 27 square inches: 27 in x 1 in or 3 in x 9

in

b) Student correctly calculates the perimeter of the specific area they

chose in part a:

If chose 27 x 1 rectangle then the perimeter = 27 + 27 + 1 +

1 = 56 inches

If chose 3 x 9 rectangle then the perimeter = 3 + 3+ 9 + 9 =

24 inches

1

1

2

5. 4.OA.3, 4NBT.5

a) Student shows any correct work to finds the total number of cans =

498

b) Student shows correct work to find three times as many as 310

1

2

4

and explains their thinking (275 x3 = 825) or 275 + 275 + 275 =

825 or uses partial sums or partial products to solve and writes a

correct corresponding equation

c) Student shows correct work and writes a correct multiplication

equation to calculate such as 24 x 13 = 312

1

Total Points 17 17

Novice Apprentice Practitioner Expert

0 – 4 points 5 – 9 points 10 – 14 points 15 – 17 points

Grade 4 Unit 3: Final Performance Assessment: Recycling Water Bottles

1. P.S. 49 is recycling water bottles for an art project. This week, Mary and Jose collected water

bottles. a) Mary collected 7 more bottles than his friend Jose. Jose collected 4 bottles. How many bottles

did Jose collect? Solve using an equation

b) The following week, Jose collected 6 bottles. Mary collected 9 times as many bottles as Jose. How many bottles did Mary collect? Solve using an equation

c) Use numbers and words to explain how questions (a) and (b) are similar and different

2. Mark is helping Ms. Smith arrange the bottles they collected. There are 30 large bottles and 43 small bottles. a) Show three different ways that Luis can arrange all 30 large bottles into equal size rows and

columns. Write a multiplication expression for each diagram.

Expression:_____________________

Expression:_____________________

Expression:_____________________

b) How many different ways can Mark arrange all 43 small bottles into equal size rows and columns? Write a multiplication expression that represents your answer

c) There are a total 73 bottles. How can you tell if the number is prime or composite? Explain

your mathematical thinking.

3. Class 5B collected 49 bottles. The students will be arranging them in crates. Each crate fits 12 bottles. How many crates does the class need? a) Draw a diagram to help solve the problem

b) Write an equation using a symbol for the unknown variable.

c) How do you know your answer is correct? Explain ___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

4. The school is thinking about using larger crates to arrange the bottles. The area of the bottom of the crate is 36 square inches.

a) What could be the dimensions (length and width) of the bottom of the crate? Show your work

b) What is the perimeter of the bottom of the crate? Show your work

5. (4OA3, 4NBT5) The table below shows the total number of bottles collected in each grade

Grades Total Bottles Collected

Grade 4 112

Grade 5 83

Grade 6 310

a) What is the total number of bottles collected? Show your work.

b) Grade 3 collected three times as many bottles than Grade 6. How many bottles did Grade 3 collect?

Show your work. Explain your mathematical thinking Write an equation: ____________________________

c) Each student in Class 5A collected 14 bottles. There 32 students in this class. How many bottles did they collect? Show your work.

Multiplication Equation:_________________________

Grade 4 Unit 3

Final Assessment Scoring Guide

Recycling Water Bottles Scoring Guide Points Section

Points

1. 4.OA.1, 4.OA.2

a) Student writes a correct equation and solves correctly

7 + 4 = 11 or 4 + 7 = 11

b) Student writes a correct equation and solves correctly

6 x 9 = 54 or 9 x 6 = 54

c) Similar: both are comparisons but “a” is an addition (additive)

comparison and “b” is a multiplication (multiplicative) comparison

1

1

1

3

2. 4.OA.4

a) Student draws THREE diagrams showing equal size rows and

columns and writes the corresponding multiplication expressions: 1 x

30, 6 x 5, 3 x 10 (note: the standard calls for finding ALL factor pairs

for a given whole number)

b) Student correctly answers that there is only one way to arrange and

writes the correct corresponding multiplication expression

1 x 43 or 43 x 1

c) Student correctly answers that because there is only one way to

arrange 73 cans into equal rows and columns or because there are

only 2 factors (1 factor pair) of 1 and itself – I know it is a prime

number

3

1

1

5

3. 4OA3, 4.NBT.6

a) Student must draw a diagram that represents the 12 cans into one

shoebox and a total of 49 cans for a total of 5 crates

b) Student correctly writes the equation 49 ÷ 12 = x or 49/12 = x or 12x =

49

c) Student can explain using any variety of reasons – can reference the

diagram, can explain that there is 1 leftover so they will need a fifth

box, etc

1

1

1

3

4. 4MD3, 4OA4

a) Student must draw a diagram that shows the dimensions of either

rectangle with an area of 36 square inches: 36 in x 1 in or 3 in x 12 or

2 x 18

b) Student correctly calculates the perimeter of the specific area they

chose in part a:

If chose 36 x 1 rectangle then the perimeter = 36 + 36 + 1 +

1 = 74 inches

If chose 3 x 12 rectangle then the perimeter = 3 + 3+ 12 +

12 = 30 inches

If chose 2 x 18 rectangle then the perimeter = 2 + 2 + 18 +

18 = 40 inches

1

1

2

5. 4.OA.3, 4NBT.5

a) Student shows any correct work to finds the total number of cans =

498

b) Student shows correct work to find three times as many as 310 and

explains their thinking (310 x3 = 930) or 310 + 310 + 310 = 930 or

uses partial sums or partial products to solve and writes a correct

corresponding equation

c) Student shows correct work and writes a correct multiplication

equation to calculate such as 14 x 32 = 448

1

2

1

4

Total Points 16 16

Novice Apprentice Practitioner Expert

0 – 4 points 5 – 9 points 10 – 13 points 14 – 16 points