THIRD GRADE > 3.OA.3 TEACHER GUIDE

TEACHER GUIDE TO CLARIFICATION

3.OA.3            ARRAYS AND DRAWINGS           

COHERENCE AND CONNECTIONS           

CLASSROOM RESOURCE             HOT QUESTIONS            ADDITIONAL RESOURCES

3.OA.3

 

Multiplication and Division with Arrays and Drawings 

 

3.OA.3 Use Multiplication and division within 100 to solve word problems in situations involving equal groups, arrays and measurement quantities, e.g., by using drawings with a symbol for the unknown number to represent the problem

Arrays and Drawings

 

Utilizing Arrays and Drawings to multiply and divide. These strategies will build the foundation needed to understand what happens to numbers when multiplied or divided. Students will be able to reason through the problems visually and then mentally. 

 

Multiplication word problem with group size unknown

 

Problem 1 Array

There are a total of 32 books. Each book shelf holds 8 books. How many shelves are there?

8 x ? = 32

or

Problem 1 Drawing 

 

Students can draw each shelf with 8 books on it. 

An Array can look like pictures or drawings in rows and columns or in a connected grid that shows thenumber of items in each row or column.

A student may know that 8 books on 3 shelves equals 24, 24 + 24=48

I need one more shelf 48 + 8=56. Therefore, the books would fill 7 shelves. There are multiple ways a student can draw the picture. Make sure students have an opportunity to explain their thinking. 

Problem 2 Multiplication Problem with an Array Area Model

 

13 x 6 = ?

Students can actually see the total with this representation. This visually representation is

setting the foundation for what happens when a number is multiplied by another. Students will

be able to understand the relationship between multiplication and addition.

 

Problem 2 Multiplication Problem with an Array Area Model 

 

13 x 6 = ? (same problem represented a little different) 

 

As the numbers get bigger, drawing each individual object or grid in its entirety might take too

much time. Students can make a box with estimated divisions.

Division word problem with the number of groups unknown

 

Problem 3 Drawing

 

I have 36 chairs and 4 tables. How many people can sit at the tables? 9 people

When answering word problem students must label their work correctly. (MP 6) 

Watch this video to see how to divide using an Area Array Model

 

In grade 4, the standard algorithm for multiplication will be introduced (4.NBT.4) Example of 4th grade problem 529 ÷ 30 using an array http://www.youtube.com/watch?v=u7M7G9bnQy0

Problem 4 Area Array Model Group Size unknown

 

A teacher has 92 pencils for each group of 4 students. How many pencils will each student receive? 

                                        4 x ? = 92        or           92 ÷ 4 = ?

 

92 ÷ 4 = ? With a problem like this, students might find it daunting to draw pencils in groups of 4 until they reach 92. We can use an area model array. This is essentially the reverse of the multiplication array. We start with a total area or the dividend. We know one of the side lengths of the area, which is the divisor which usually goes on the bottom of the area model as a known number. The quotient is then determined by the student in partial products type procedure. 

 

92 is the total area of this array. But there are partitions that can be made to help us determine the group unknowns. I will start on the bottom and draw a line in the array model that will represent 10 groups of 4 which is 40 pencils. But I need 92 so I can draw another line to create 10 groups of 40. This brings me to 80 pencils so 12 are left. This means the top area is 12 pencils. I had to use 23 groups of 4 to make a total of 92 pencils. Each student in the group of 4 would get 23 pencils. 

3 X 4=12 pencils

10 X 4=40 pencils

10 X 4=40 pencils

4 groups of students 

Same Problem different way to think about it. This drawing has also taken out all of the grid lines and has partitioned according to the partial areas. 

Problem 5 Area Array Model Number of groups unknown

 

My friend is having a picnic. She has 69 graham crackers. Each student will receive 3 crackers. How many students will be going on the picnic? After solving with an area array model I can see that 23 students will be going to the picnic.

Coherence and Connections: Need to Know

Grade Below

4.OA.1

4.OA.2

4.NF.4c

Grade Level

3.OA.1

3.OA.2

3.OA.3

3.OA.4

3.OA.7

3.OA.8

Grade Above

Evidence

Statement Key

Evidence Statement Text

Clarifications

MP

3.OA.3-1

 

Use multiplication within 100 (both factors less than or equal to 10) to solve word problems in situations involving equal groups, arrays, or area, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. 

i) All products come from the harder three quadrants of the times table (a × b where a > 5 and/or b > 5).

ii) 50% of tasks involve multiplying to find the total number (equal groups, arrays); 50% involve multiplying to find the area.

iii) For more information see CCSS Table 2, p. 89 and the OA Progression. 

1,4

3.OA.3-2

 

Use multiplication within 100 (both factors less than or equal to 10) to solve word problems in situations involving measurement quantities other than area, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. 

i) All products come from the harder three quadrants of the times table (a × b where a > 5 and/or b > 5).

ii) Tasks involve multiplying to find a total measure (other than area).

iii) For more information see CCSS Table 2, p. 89 and the OA Progression. 

1,4

3.OA.3-3

 

Use division within 100 (quotients related to products having both factors less than or equal to 10) to solve word problems in situations involving equal groups, arrays, or area, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. 

i) All quotients are related to products from the harder three quadrants of the times table (a × b where a > 5 and/or b > 5).

ii) A third of tasks involve dividing to find the number in each equal group or in each equal row/column of an array; a third of tasks involve dividing to find the number of equal groups or the number of equal rows/columns of an array; a third of tasks involve dividing an area by a side length to find an unknown side length.

iii) For more information see CCSS Table 2, p. 89 and the OA Progression.Use multiplication within 100 (both factors less than or equal to 10) to solve word problems in situations involving measurement quantities other than area, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. 

1,4

3.OA.3-4

 

Use division within 100 (quotients related to products having both factors less than or equal to 10) to solve word problems in situations involving measurement quantities other than area, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. 

i) All quotients are related to products from the harder three quadrants of the times table (a × b where a > 5 and/or b > 5). ii) 50% of tasks involve finding the number of equal pieces; 50% involve finding the measure of each piece.

iii) For more information see CCSS Table 2, p. 89 and the OA Progression. Use division within 100 (quotients related to products having both factors less than or equal to 10) to solve word problems in situations involving equal groups, arrays, or area, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. 

1,4

Illinois Assessment of Readiness Mathematics Evidence Tables. Retrieved from:

https://www.isbe.net/Documents/IAR-Grade-3-Math-Evidence-State.pdf

Take a look at the clarification statements. This give an idea of what students will need to know and be able to do. It also gives a suggestion to the percentages of different tasks on the assessment. 

It is important to establish the foundation of arrays in 3rd grade. This strategy is used with larger numbers in 4th grade. This standard refers to Table 2 in the glossary of the standards.

Problems should be presented with unknown factors along with unknown products. 

 

See 4th grade division video

Multiplication and Division problems should be written both horizontally and vertically. Students can then apply the strategy they are comfortable with.

 
 
 
 
 
 
 
Classroom Resources
 

3.OA.3 Daily Discourse power point with multiple multiplication and division word problems and a variety of strategies 

HOT Questions

3.OA.3 Use Multiplication and division within 100 to solve word problems in situations involving equal groups, arrays and measurement quantities, e.g., by using drawings with a symbol for the unknown number to represent the problem.

 

Problem 1

I have 18 cookies that I would like to split into 6 boxes. Be able to explain your answer and strategy used. 

 

Problem 2

My friend drew an area model array to help him multiply. What would the missing numbers be?

What is the total and solve for a and b?

18 x 4 = ? a = ? b = ?

10               +               8 

a

40

b

Answers

 

Total = 72

 

a=4

b=32

Problem 3

I have 45 beads and I would like to make 3 necklaces. How many beads would be on each

necklace? What would the missing number (a) be? 

45 beads

a

5

30 beads

15 beads

3 necklaces

Answers

 

15 beads on

each necklace

 

a=10

3.OA.3            ARRAYS AND DRAWINGS           

COHERENCE AND CONNECTIONS           

CLASSROOM RESOURCE             HOT QUESTIONS            ADDITIONAL RESOURCES

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