4.OA.1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 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 comparisons 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.
Anchor Standard/Mathematical Practice(s)
MP.1. Make sense of problems and persevere in solving them. MP.2. Reason abstractly and quantitatively. MP.4. Model with mathematics. MP.5. Use appropriate tools strategically. MP.6. Attend to precision. MP.7. Look for and make use of structure.
Information Technology Standard
Use technology tools and skills to reinforce classroom concepts and activities.
A multiplicative comparison is a situation in which one quantity is multiplied by a specified number to get
another quantity (e.g., “a is n times as much as b”). Students should be able to identify and verbalize which quantity is being multiplied and which number tells how many times. Students should be given opportunities to write and identify equations and statements for multiplicative comparisons.
Example:
5 x 8 = 40. Sally is five years old. Her mom is eight times older. How old is Sally’s Mom?
5 x 5 = 25
Sally has five times as many pencils as Mary. If Sally has 5 pencils, how many does Mary have? This standard calls for students to translate comparative situations into equations with an unknown and solve. Students need many opportunities to solve contextual problems. Refer to Glossary, Table 2(page 89) For more examples (table included at the end of this document for your convenience) Examples: Unknown Product: A blue scarf costs $3. A red scarf costs 6 times as much. How much does the red scarf cost? (3 x 6 = p). Group Size Unknown: A book costs $18. That is 3 times more than a DVD. How much does a DVD cost? (18 ÷ p = 3 or 3 x p = 18). Number of Groups Unknown: A red scarf costs $18. A blue scarf costs $6. How many times as much does the red scarf cost compared to the blue scarf? (18 ÷ 6 = p or 6 x p = 18).
When distinguishing multiplicative comparison from additive comparison, students should note that
•additive comparisons focus on the difference between two quantities (e.g., Deb has 3 apples and Karen has 5 apples. How many more apples does Karen have?). A simple way to remember this is, “How many more?” •multiplicative comparisons focus on comparing two quantities by showing that one quantity is a specified number of times larger or smaller than the other (e.g., Deb ran 3 miles. Karen ran 5 times as many miles as Deb. How many miles did Karen run?). A simple way to remember this is “How many times as much?” or “How many times as many?”
The focus in this standard is to have students use and discuss various strategies. It refers to estimation strategies,
including using compatible numbers (numbers that sum to 10 or 100) or rounding. Problems should be structured so that all acceptable estimation strategies will arrive at a reasonable answer. Students need many opportunities solving multistep story problems using all four operations.
Example:
On a vacation, your family travels 267 miles on the first day, 194 miles on the second day and 34 miles on the third day. How many miles did they travel total?
Some typical estimation strategies for this problem:
Student 1 I first thought about 267 and 34. I noticed that their sum is about 300. Then I knew that 194 is close to 200. When I put 300 and 200 together, I get 500. Student 2 I first thought about 194. It is really close to 200. I also have 2 hundreds in 267. That gives me a total of 4 hundreds. Then I have 67 in 267 and the 34. When I put 67 and 34 together that is really close to 100. When I add that hundred to the 4 hundreds that I already had, I end up with 500. Student 3 I rounded 267 to 300. I rounded 194 to 200. I rounded 34 to 30. When I added 300, 200 and 30, I know my answer will be about 530.
The assessment of estimation strategies should only have one reasonable answer (500 or 530), or a range
(between 500 and 550). Problems will be structured so that all acceptable estimation strategies will arrive at a reasonable answer.
Example 2:
Your class is collecting bottled water for a service project. The goal is to collect 300 bottles of water. On the first day, Max brings in 3 packs with 6 bottles in each container. Sarah wheels in 6 packs with 6 bottles in each container. About how many bottles of water still need to be collected?
This standard references interpreting remainders. Remainders should be put into context for interpretation.
ways to address remainders: •Remain as a left over •Partitioned into fractions or decimals •Discarded leaving only the whole number answer •Increase the whole number answer up one •Round to the nearest whole number for an approximate result
Example:
Write different word problems involving 44 / 6 = ? where the answers are best represented as:
Problem A: 7
Problem B: 7 r 2
Problem C: 8
Problem D: 7 or 8
Problem E: 7 2/6
possible solutions:
Problem
A: 7. Mary had 44 pencils. Six pencils fit into each of her pencil pouches. How many pouches did she fill? 44 ÷ 6 = p; p = 7 r 2. Mary can fill 7 pouches completely.
Problem
B: 7 r 2. Mary had 44 pencils. Six pencils fit into each of her pencil pouches. How many pouches could she fill and how many pencils would she have left? 44 ÷ 6 = p; p = 7 r 2; Mary can fill 7 pouches and have 2 left over.
Problem C. 8.
Mary had 44 pencils. Six pencils fit into each of her pencil pouches. What would the fewest number of pouches she would need in order to hold all of her pencils? 44 ÷ 6 = p; p = 7 r 2; Mary can needs 8 pouches to hold all of the pencils.
Problem
D: 7 or 8. Mary had 44 pencils. She divided them equally among her friends before giving one of the leftovers to each of her friends. How many pencils could her friends have received? 44 ÷ 6 = p; p = 7 r 2; Some of her friends received 7 pencils. Two friends received 8 pencils.
Problem
E: 7 2/6 Mary had 44 pencils and put six pencils in each pouch. What fraction represents the number of pouches that Mary filled? 44 ÷ 6 = p; p = 7 2/6
Example:
There are 128 students going on a field trip. If each bus held 30 students, how many buses are needed? (128 ÷ 30 = b; b = 4 R 8; They will need 5 buses because 4 busses would not hold all of the students). Students need to realize in problems, such as the example above, that an extra bus is needed for the 8 students that are left over.
Estimation skills include identifying when estimation is appropriate, determining the level of accuracy needed,
selecting the appropriate method of estimation, and verifying solutions or determining the reasonableness of situations using various estimation strategies. Estimation strategies include, but are not limited to: •front-end estimation with adjusting (using the highest place value and estimating from the front end, making adjustments to the estimate by taking into account the remaining amounts), •clustering around an average (when the values are close together an average value is selected and multiplied by the number of values to determine an estimate), •rounding and adjusting (students round down or round up and then adjust their estimate depending on how much the rounding affected the original values), •using friendly or compatible numbers such as factors (students seek to fit numbers together - e.g., rounding to factors and grouping numbers together that have round sums like 100 or 1000), •using benchmark numbers that are easy to compute (students select close whole numbers for fractions or decimals to determine an estimate). I can... I can write an equation for my multiplication word problem. I can use algebraic thinking to solve word problems involving multiplication and division. I can use the 4 operations to solve multi step word problems containing whole numbers. I can interpret the remainder in a division word problem. I can make sure my answer makes sense.
Evaluate 8 x 6, 5 x 9, 7 x 3, etc. Write the equations for each multiplication problem using the Commutative Property of Multiplication (e.g., 8 x 6 = 6 x 8). Write and then solve the given equation using another method. Use a verbal statement to explain the chosen method.
Pedro has invited 8 of his friends to a summer party. He asked each of them to bring 7 pieces of candy. Create a representation of the total number of candy pieces the friends will share. Write the equation that represents the illustration you created. Solve for the answer.
Over the summer, Raul read 8 books. Natalia read 4 times as many books. How many books did Natalia read? Draw a picture or create a model of the problem, write an equation with a symbol for the unknown variable, and solve.
Breaking apart-decomposing 25 x 8 = 20 x 8 = 160 5 x 8 = 40 160+40=200
Array Model/Area Model 17 x 5
10
7
5
50
35
50 + 35 85 Lattice Math Choice Boards Use playing cards for multiplication war Post words with visuals on word wall and in vocabulary notebooks. Use Frayer Models and foldables
Enrichment:
Students create and solve real world word problems. Learning Logs In small groups create a board game involving multiplication
Common Core Standards
4.OA.1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 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 comparisons 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.
Anchor Standard/Mathematical Practice(s)
MP.1. Make sense of problems and persevere in solving them.MP.2. Reason abstractly and quantitatively.
MP.4. Model with mathematics.
MP.5. Use appropriate tools strategically.
MP.6. Attend to precision.
MP.7. Look for and make use of structure.
Information Technology Standard
Use technology tools and skills to reinforce classroom concepts and activities.Revised Bloom's Level of thinking
RememberingUnderstanding
Applying
Analyzing
Evaluating
Creating
Learning Target/Task Analysis
A multiplicative comparison is a situation in which one quantity is multiplied by a specified number to get
another quantity (e.g., “a is n times as much as b”). Students should be able to identify and verbalize which
quantity is being multiplied and which number tells how many times.
Students should be given opportunities to write and identify equations and statements for multiplicative
comparisons.
Example:
5 x 8 = 40.
Sally is five years old. Her mom is eight times older. How old is Sally’s Mom?
5 x 5 = 25
Sally has five times as many pencils as Mary. If Sally has 5 pencils, how many does Mary have?
This standard calls for students to translate comparative situations into equations with an unknown and solve.
Students need many opportunities to solve contextual problems. Refer to Glossary, Table 2(page 89)
For more examples (table included at the end of this document for your convenience)
Examples:
Unknown Product: A blue scarf costs $3. A red scarf costs 6 times as much. How much does the red scarf cost?
(3 x 6 = p).
Group Size Unknown: A book costs $18. That is 3 times more than a DVD. How much does a DVD cost?
(18 ÷ p = 3 or 3 x p = 18).
Number of Groups Unknown: A red scarf costs $18. A blue scarf costs $6. How many times as much does the red
scarf cost compared to the blue scarf? (18 ÷ 6 = p or 6 x p = 18).
When distinguishing multiplicative comparison from additive comparison, students should note that
•additive comparisons focus on the difference between two quantities (e.g., Deb has 3 apples and Karen
has 5 apples. How many more apples does Karen have?). A simple way to remember this is, “How many
more?”
•multiplicative comparisons focus on comparing two quantities by showing that one quantity is a specified number of times larger or smaller than the other (e.g., Deb ran 3 miles. Karen ran 5 times as many miles
as Deb. How many miles did Karen run?). A simple way to remember this is “How many times as much?” or “How many times as many?”
The focus in this standard is to have students use and discuss various strategies. It refers to estimation strategies,
including using compatible numbers (numbers that sum to 10 or 100) or rounding. Problems should be structured
so that all acceptable estimation strategies will arrive at a reasonable answer. Students need many opportunities
solving multistep story problems using all four operations.
Example:
On a vacation, your family travels 267 miles on the first day, 194 miles on the second day and 34 miles on the
third day. How many miles did they travel total?
Some typical estimation strategies for this problem:
Student 1
I first thought about 267 and 34. I noticed that their sum is about 300. Then I knew that 194 is close to 200. When I put 300 and 200
together, I get 500.
Student 2
I first thought about 194. It is really close to 200. I also have 2 hundreds in 267. That gives me a total of 4 hundreds. Then I have 67 in 267 and the 34.
When I put 67 and 34 together that is really close to 100. When I add that hundred to the 4 hundreds that I already had, I end up with 500.
Student 3
I rounded 267 to 300. I rounded 194 to 200. I rounded 34 to 30. When I added 300, 200 and 30, I know my answer will be about 530.
The assessment of estimation strategies should only have one reasonable answer (500 or 530), or a range
(between 500 and 550). Problems will be structured so that all acceptable estimation strategies will arrive at a
reasonable answer.
Example 2:
Your class is collecting bottled water for a service project. The goal is to collect 300 bottles of water. On the first
day, Max brings in 3 packs with 6 bottles in each container. Sarah wheels in 6 packs with 6 bottles in each
container. About how many bottles of water still need to be collected?
This standard references interpreting remainders. Remainders should be put into context for interpretation.
ways to address remainders:
•Remain as a left over
•Partitioned into fractions or decimals
•Discarded leaving only the whole number answer
•Increase the whole number answer up one
•Round to the nearest whole number for an approximate result
Example:
Write different word problems involving 44 / 6 = ?
where the answers are best represented as:
Problem A: 7
Problem B: 7 r 2
Problem C: 8
Problem D: 7 or 8
Problem E: 7 2/6
possible solutions:
Problem
A: 7.
Mary had 44 pencils. Six pencils fit into each of her pencil pouches. How many pouches did she fill? 44 ÷ 6 = p; p = 7 r 2. Mary can fill 7 pouches completely.
Problem
B: 7 r 2.
Mary had 44 pencils. Six pencils fit into each of her pencil pouches. How many pouches could she fill and how many pencils would she have left?
44 ÷ 6 = p; p = 7 r 2; Mary can fill 7 pouches and have 2 left over.
Problem
C. 8.
Mary had 44 pencils. Six pencils fit into each of her pencil pouches. What would the fewest number of pouches she would need in order to hold all of her pencils?
44 ÷ 6 = p; p = 7 r 2; Mary can needs 8 pouches to hold all of the pencils.
Problem
D: 7 or 8.
Mary had 44 pencils. She divided them equally among her friends before giving one of the leftovers to each of her friends. How many pencils could her friends have received? 44 ÷ 6 = p; p = 7 r 2; Some of her friends received 7 pencils. Two friends received 8 pencils.
Problem
E: 7 2/6
Mary had 44 pencils and put six pencils in each pouch. What fraction represents the number of pouches that Mary filled? 44 ÷ 6 = p; p = 7 2/6
Example:
There are 128 students going on a field trip. If each bus held 30 students, how many buses are needed?
(128 ÷ 30 = b; b = 4 R 8; They will need 5 buses because 4 busses would not hold all of the students).
Students need to realize in problems, such as the example above, that an extra bus is needed for the 8 students
that are left over.
Estimation skills include identifying when estimation is appropriate, determining the level of accuracy needed,
selecting the appropriate method of estimation, and verifying solutions or determining the reasonableness of
situations using various estimation strategies. Estimation strategies include, but are not limited to:
•front-end estimation with adjusting (using the highest place value and estimating from the front end,
making adjustments to the estimate by taking into account the remaining amounts),
•clustering around an average (when the values are close together an average value is selected and
multiplied by the number of values to determine an estimate),
•rounding and adjusting (students round down or round up and then adjust their estimate depending on
how much the rounding affected the original values),
•using friendly or compatible numbers such as factors (students seek to fit numbers together - e.g.,
rounding to factors and grouping numbers together that have round sums like 100 or 1000),
•using benchmark numbers that are easy to compute (students select close whole numbers for fractions or decimals to determine an estimate).
I can...
I can write an equation for my multiplication word problem.
I can use algebraic thinking to solve word problems involving multiplication and division.
I can use the 4 operations to solve multi step word problems containing whole numbers.
I can interpret the remainder in a division word problem.
I can make sure my answer makes sense.
Essential Vocabulary
multiplication equation, multiply, divide, unknown, algebraic thinking, equation, remainders, reasonableness, mental computation, estimation strategies, rounding
Sample Assessments
ticket out the doorEvaluate 8 x 6, 5 x 9, 7 x 3, etc. Write the equations for each multiplication problem using the Commutative Property of Multiplication (e.g., 8 x 6 = 6 x 8). Write and then solve the given equation using another method. Use a verbal statement to explain the chosen method.
Pedro has invited 8 of his friends to a summer party. He asked each of them to bring 7 pieces of candy. Create a representation of the total number of candy pieces the friends will share. Write the equation that represents the illustration you created. Solve for the answer.
Over the summer, Raul read 8 books. Natalia read 4 times as many books. How many books did Natalia read? Draw a picture or create a model of the problem, write an equation with a symbol for the unknown variable, and solve.
Differentiation
Intervention:
http://illuminations.nctm.org/ActivityDetail.aspx?ID=155 multiplication facts reviewUse base ten blocks
Breaking apart-decomposing
25 x 8 =
20 x 8 = 160
5 x 8 = 40
160+40=200
Array Model/Area Model
17 x 5
85
Lattice Math
Choice Boards
Use playing cards for multiplication war
Post words with visuals on word wall and in vocabulary notebooks.
Use Frayer Models and foldables
Enrichment:
Students create and solve real world word problems.Learning Logs
In small groups create a board game involving multiplication
Instructional Resources
http://www.quia.com/mathjourney.htmlhttp://www.coolmath-games.com/numbermonster/mult25.htm
http://classroom.jc-schools.net/basic/mathasmd.html
Notes and Additional Information