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ArithmeticAddition and subtraction

Properties of equalityHere you will learn how to multiply multi digit numbers with strategies, such as the relationship between addition and subtraction, the properties of operations, and how to use the standard algorithm.

Students will first learn about multiplying multi digit numbers in 4 th grade and learn to use the standard algorithm to multiply in 5 th grade.

**Multiplying multi digit numbers** is multiplying large numbers together and finding their product.

When first working with multi-digit multiplication, the focus is on using strategies including place value understanding, properties of operations and the relationship between multiplication and division.

For example,

20 \times 61= \; ?

- I know that 10 \times 61 = 610, so I can double that to find 20 \times 61.

610 + 610 = 1,220

- I know that 61 = 30 + 30 + 1.

Since 30 \times 20 = 600 and 1 \times 20 = 20. This means…

\begin{aligned} & 20 \times 61 \\ & =20 \times(30+30+1) \\ & =600+600+20 \\ & =1,220 \end{aligned}

- I can use an area model to solve.

Notice there are many ways to solve the same problem – using partial products, using the distributive property and area models. Knowing the names of each strategy is not important, but understanding why they work and making connections between them is.

After students have had sufficient practice understanding division with mostly single-digit numbers in 3 rd grade and multi-digit whole numbers in 4 th, the standard algorithm is taught in 5 th grade.

For example,

20 \times 61= \; ?

Start by writing the column multiplication equation. Leave an empty row at the top.

First, multiply the top number by the ones place in the bottom number (multiplier).

This shows that 20 \times 1=20

Then multiply the top number by the tens place of the bottom number (multiplier).

Since the 6 in 61 represents 60, we place a 0 in the ones position as a place holder.

This shows that 60 \times 20=1,200

Notice that since multiplying 6 \times 2=12 produces 2 digits in the product, this step requires regrouping. You record the 2 and regroup the 1 to the next position – in this case the thousands.

Finally, add the partial products for the final answer.

So 20 \times 61=1,220.

Note: This page does not cover multiplication with decimals. However, students use strategies in fifth grade to multiply decimals up to hundredths and use the standard algorithm in 6th grade to multiply all decimals.

**See also:** Multiplying and dividing decimals

How does this apply to 4 th grade math and 5 th grade math?

**Grade 4: Numbers and Operations in Base 10 (4.NBT.B.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.

**Grade 5: Numbers and Operations in Base 10 (5.NBT.B.5)**Fluently multiply multi-digit whole numbers using the standard algorithm.

Use this quiz to check your grade 4, 5 and 7 students’ understanding of multiplication and division. 10+ questions with answers covering a range of 4, 5 and 7 grade multiplication and division topics to identify areas of strength and support!

DOWNLOAD FREEUse this quiz to check your grade 4, 5 and 7 students’ understanding of multiplication and division. 10+ questions with answers covering a range of 4, 5 and 7 grade multiplication and division topics to identify areas of strength and support!

DOWNLOAD FREEIn order to multiply multi digit numbers with strategies:

**Write the multiplication equation.****Choose a solving strategy.****Solve the equation.**

Solve 19 \times 8

**Write the multiplication equation.**

19 \times 8= \; ?

2**Choose a solving strategy.**

19 \times 8 is one group of 8 less than 20 \times 8. You can use this related fact to solve.

3**Solve the equation.**

20 \times 8=160 and 160-8=152.

So 19 \times 8=152.

Solve 238 \times 6.

**Write the multiplication equation.**

238 \times 6= \; ?

**Choose a solving strategy.**

You can use an area model to solve.

**Solve the equation.**

So 238 \times 6=1,428.

There are 81 classroom novel sets. Each novel set has 34 books. How many books are there in all the sets?

**Write the multiplication equation.**

There are 81 groups of 34, which is written as…

34 \times 81= \; ?

**Choose a solving strategy.**

You can use partial products to solve.

**Solve the equation.**

34 \times 81

=(4+30) \times 81 \quad **Break 34 up into 4 + 30 and multiply both by 81

=4 \times 81+30 \times 81

=324+30 \times(80+1) \quad **Break 81 up into 80 + 1 and multiply both by 30

=324+2,400+30

=2,754

So 34 \times 81=2,754

In order to multiply multi digit numbers with the standard algorithm:

**Write the column multiplication equation.****Multiply the top number by the ones position of the bottom number (multiplier) – recording the partial product.****Repeat for each position in the multiplier.****Add all the partial products together.**

Solve 34 \times 7 using the standard algorithm.

**Write the column multiplication equation.**

Start by writing the column multiplication equation. Leave an empty row at the top.

**Multiply the top number by the ones position of the bottom number (multiplier) – recording the partial product.**

First, multiply the top number by the ones place in the bottom number (multiplier).

Notice that since multiplying 7 \times 4=28 produces 2 digits in the product, this step requires regrouping. You record the ones digit and regroup the tens digit to the next position.

Notice that we add the regrouped 2 after multiplying: 3 \times 7=21 and 21+2=23.

**Repeat for each position in the multiplier.**

7 is the only digit in the multiplier – so there is nothing to repeat.

**Add all the partial products together.**

There are no partial products. The final answer is shown.

34 \times 7=238.

Solve 281 \times 5 using the standard algorithm.

**Write the column multiplication equation.**

Start by writing the column multiplication equation. Leave an empty row at the top.

First, multiply the top number by the ones place in the bottom number (multiplier).

**Repeat for each position in the multiplier.**

5 is the only digit in the multiplier – so there is nothing to repeat.

**Add all the partial products together.**

There are no partial products. The final answer is shown.

281 \times 5=1,405.

Solve 37 \times 19 using the standard algorithm.

**Write the column multiplication equation.**

Start by writing the column multiplication equation. Leave an empty row at the top.

First, multiply the top number by the ones place in the bottom number (multiplier).

This shows that 9 \times 37=333.

**Repeat for each position in the multiplier.**

Next, multiply the top number by the tens place in the bottom number (multiplier).

Since the 1 in 17 is in the tens place, remember to put 0 as a place holder in the ones column.

This shows that 10 \times 37=370.

**Add all the partial products together.**

The final product is 37 \times 19=703.

- Before multiplying with larger numbers, students should have had plenty of opportunities to learn multiplication of 1- digit numbers. Preferably, students will have memorized their multiplication facts, before moving on to 2- digit multiplication and beyond, but this is not a requirement. You can provide a multiplication table to students who are still mastering their basic facts.

- Emphasize students making sense of multiplication problems over speed related activities. This includes encouraging students to show their work, solve in more than one way, compare their strategies to others, and use estimation or other strategies to consider if their answer is reasonable.

- While math worksheets and multi-digit multiplication worksheets can be useful for this skill, make sure to expose students to all types of multiplication problems; ones that provide many different contexts and opportunities to solve in different ways. This includes using a mixture of word problems, projects, real world scenarios, calculator activities, etc. throughout your lesson plans.

- As students are learning the algorithm, provide them with a step by step guide and/or graphic organizer that explains each part. This gives them something to refer back to and can help them make sense of the algorithm.

**Thinking only one multiplication strategy or algorithm will work**When using strategies, students can solve problems with related facts, properties of numbers, partial products, area models and more. Additionally, even though one standard algorithm is taught, there are others. Some students may be familiar with the Lattice algorithm or others.

**Incorrectly adding partial products**When learning to multiply large numbers, many students use the partial products method. For this method to be successful, students need to keep track of all the partial products and add them together at the end. If a student’s work is messy or incomplete, it can lead to mistakes.

**Not using place value when solving**

Place value should always be considered when solving – no matter the strategy or algorithm. This helps prevent mistakes and can be used to identify whether or not an answer is reasonable.

**Thinking that the top number has to be larger than the bottom**

For example,

34 \times 815 can be set up either way:

While the setup on the right is the most common, both will work.

- Multiplication and division
- Multiplicative comparison
- Dividing multi digit numbers
- Multiplying and dividing integers
- Multiplying and dividing rational numbers
- Understanding multiplication
- Multiplication and division within 100
- Long division
- Understanding division

1. Solve 88 \times 3.

48

270

264

244

You can use a related fact to solve.

88 \times 3 is two groups of 3 less than 90 \times 3.

90 \times 3=270

and

270-3-3=264.

So 88 \times 3=264.

2. Solve 801 \times 9.

7,209

81

729

8,100

Break up 801 into parts \rightarrow 801=800+1.

Since 800 \times 9=7,200 and 1 \times 9=9. This means…

\begin{aligned} & 801 \times 9 \\\\ & =(800+1) \times 9 \\\\ & =(800 \times 9)+(1 \times 9) \\\\ & =7,200+9 \\\\ & =7,209 \end{aligned}

3. Each recipe calls for 18 ounces of fruit. Jeylani wants to make 33 recipes. How many ounces of fruit does she need?

54

594

297

300

Use an area model to solve.

4. Solve 76 \times 4 with the algorithm.

284

2,824

52

304

Notice that since multiplying 6 \times 4=24 produces 2 digits in the product, this step requires regrouping.

Notice that we added the regrouped 2 after multiplying: 4 \times 8=28 and 28 + 2 = 30.

5. Solve 925 \times 4 with the algorithm.

2,780

3,700

3,600

3,780

6. An English teacher requires his students to read 45 minutes each night. How many total minutes will each student have read after 41 nights?

1,845

1,645

225

1,800

In third grade, students work with multiplying one-digit numbers within 100. Their work with multiplication is directly connected to their work with division.

Students extend their work in fourth grade to multi-digit multiplication by multiplying 2- digit numbers, 3- digit numbers and 4- digit numbers.

In 5 th grade, they learn how to solve problems with the standard algorithm. In all three grade levels, students use their reasoning and understanding of numbers and multiplication to solve.

It is important to note that there are additional standards where students focus on how to multiply by 10, 100, etc., through place value understanding. This work comes before learning how to multiply with other larger numbers.

Students begin multiplying fractions by whole numbers in 4 th grade, in both mathematical and real world contexts. Students expand to multiply fractions by fractions in 5 th grade – using strategies and algorithms.

This is also when they begin to multiply with decimals – but only using strategies and place value understanding. Students learn the algorithm to multiply with decimals in 6 th grade.

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[FREE] Common Core Practice Tests (Grades 3 to 6)

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Prepare for math tests in your state with these Grade 3 to Grade 6 practice assessments for Common Core and state equivalents.

40 multiple choice questions and detailed answers to support test prep, created by US math experts covering a range of topics!