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Using estimation in multiplying two-digit numbers

Standards

CCSS.MATH.CONTENT.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.

CCSS.MATH.CONTENT.4.OA.A.3 Solve multistep word problems posed with whole numbers and having whole-number answers using the four operation … Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Time

This lesson takes 40 minutes to an hour, depending on how long you allot your students to complete each of the 4 problem sets and whether you have students play the game.

Technology Required

Computer with a projector, Smartboard or other device for your class to watch videos or videos can be shared in Google meet or other application for remote learning. Spirit Lake: The Game can be downloaded and played on Windows or Mac computers. An iPad version will be available by Fall 2021. Schools that are part of the Growing Math project or who have a 7 Generation Games site license will have access to all of these games for students to use at home or school.

Optional Recommended Resource

Use resources like the Helping Math Multiplication Worksheet Generator to make your own worksheet. Their worksheets look very much like this one on multiplication of two-digit numbers with Easter egg examples, except with many more activities in each worksheet. Note that the link will require you to register for a free download. Registration takes about 1 minute. You’ll then receive a link to download over 100 free worksheets on math topics from addition to rational numbers. Another resource we like is the Lizard Point math worksheet generator. Both will print the worksheets as well as the answer key. Note that sites using PRINT TO Google Drive may have that feature disabled after December, 2020. However, you can still from the PRINT menu select SAVE AS PDF, download a file as PDF and then upload it to your Google Drive. The Growing Math project has no affiliation with Helping Math or Lizard Point.

Summary

You’ll need 20 math problems multiplying two-digit numbers. You can use math problems you already have or use one of the online worksheet generators. Based on research showing the effectiveness of distributed practice – that is, practicing a skill for more, shorter periods rather than one long session – we have students solve a few problems at a time, with videos in between. The lesson begins with a brief explanation of multiplying two digit numbers, followed by students solving 3-5 problems multiplying two-digit numbers. Students then watch a 3-minute video that works an example of multiplying a three-digit number by a two-digit number. They then solve 3-5 more problems, followed by another video, then more problems. We recommend giving a time frame – say, 7-10 minutes – to solve the problems rather than set number of problems because this allows you to begin and end each section of the lesson with all of the students at once.

Lesson Plan

1. Introduction to Multiplying Two-digit Numbers

Begin with an example from Helping with Math, that gives an explanation of multiplication and a variety of types of problems. Then, have the students try to solve 3 to 5 problems. You can use those included in the Google slides linked or create your own.

2. Video : Multiplication and Estimation

Estimation is one of the most practical math skills!

There are a lot of math concepts that I use regularly when writing software or computing statistics in my day job. The one skill I use all the time is estimation. (I can tell you that the ability to accurately estimate an answer is not universal.)

Example 1: Long multiplication using “math reality checks.”

In the problem in the video above, we start out by multiplying 892 x 11, using the fact that any number multiplied by 10 is just that number with an added zero. To test our answer, we round 892 to 900 and can estimate that our answer should be near to – and less than – 9,900.

Let’s say you type the wrong number in your phone, hitting the 6 instead of the 9, since these two are pretty close and you have big fingers. Now your answer is 7,612. If you have a good grasp of estimation and multiplication, that is clearly wrong. If you’re computing how much money you need to charge a customer based on the 892 hours you expect to work at $11.00 per hour, you have just lost out on over $2,000!

3. More multiplication problems

Now that you have had a little practice and a little instruction, students solve another 3-5 problems.

4. Video : Multiplication and Estimation

This video works through a problem multiplying a 3 -digit number by a two-digit number . It also gives a strategy for solving difficult problems. That is, break the problem into smaller, easier problems.

Video 2: Use addition to solve a multiplication problem.

Here, because the last digit of one number is 7 and of the other number is 2, you know that 7 x 2 = 14. So, whatever else your answer is, it has to end in a 4.

5. Follow up to the video

As a math teacher, I heard approximately 4,897,234 times from students:

WHY do I have to do so many problems of the same type?

– Almost every student I ever taught math

Explain to students that all of those facts learned, like 2 x 7 =1 4 are the basis for the commutative property of multiplication, reducing numbers to lowest terms, or solving equations by multiplying or dividing both sides by a constant, and more. You need a BASE to work from, problems that can be examples.

The fact is that the more experience you have with numbers, the more problems you solve. The more problems solved, the easier it gets.

6. Finish off by a few more math problems

Now that you’ve watched another video, it’s time for the last set of math problems.

7. Optional – Game Play

Students can Spirit Lake: The Game . Multiplying two- and three-digit numbers comes up in Level 4. We recommend allowing students to play for 15-30 minutes each session. This gives them enough time to get started but not enough time to get bored.

Assessment

Answers to the 12 -20 math problems completed by the student provide one assessment. A second assessment is in the reports for the Spirit Lake game, which show whether students answered correctly the problems in Level 4, whether they read the hint before answering the problem, and whether they were correct on the first try or had to attempt the problem more than once.

State Standards

Minnesota Math Standard 4.1.1.3 – Multiply multi-digit numbers, using efficient and generalizable procedures, based on knowledge of place value, including standard algorithms.

Minnesota Math Standard 5.1.1.4 – Solve real-world and mathematical problems requiring addition, subtraction, multiplication and division of multi-digit whole numbers. Use various strategies, including the inverse relationships between operations, the use of technology, and the context of the problem to assess the reasonableness of results.

Multiplying one-digit numbers from 6 to 9

Standards

CCSS.MATH.CONTENT.3.OA.A.1 – Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.

CCSS.MATH.CONTENT.3.OA.A.3– Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities.

CCSS.MATH.CONTENT.3.OA.A.4-Determine the unknown whole number in a multiplication or division equation relating three whole numbers.

CCSS.MATH.CONTENT.3.OA.C.7– Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

Time

Each lesson will require 20-30 minutes. With the four lesson plans, total time is approximately 1 1/4 to 2 hours spread over one to two weeks.

Technology Required

Making Camp Premium plays in any browser, so, of course, on Chromebooks. It can also be downloaded on phones or tablets and played offline by students who have limited Internet access. The teacher will need a computer, for showing to students learning from home, and a projector if showing videos in the classroom. If the classroom does not have access to a projector, the videos can be skipped. Spirit Lake: The Game can be downloaded and played on Windows or Mac computers. An iPad version will be available by Fall 2021. Schools that are part of the Growing Math project or who have a 7 Generation Games site license will have access to all of these games for students to use at home or school.

Summary

Our single-digit multiplication series is for Grade 3, although some teachers find themselves doing this in grade 4. The lesson plans with PDF links for printing and Google slide presentations are provided below. We assume you have already completed multiplication tables for 0 through 5 and for the 10s table.  While it may be tempting to do all lessons in one week, we have found it works better to spread this over a two-to three-week unit.

  • Students use visual drawings, manipulatives, and a number line to learn multiplication of one-digit numbers, coupled with their verbal explanations.
  • Students will create number sentences independently and with a partner.
  • Learning and memorizing multiplication patterns will improve later understanding of division.
  • Determine the unknown whole number in a multiplication equation relating three whole numbers.
  • Optional Brain Power activities have students write their own word problems.
  • Students play games that reinforce memory and apply multiplication in word problems.

Lesson Plans

Learning from home / hybrid learning tip

Several lessons include an activity, “Work with a partner.  Take turns creating your own number sentences and solving them.” This can be modified in three ways:

  • In a hybrid class, where students come in half of the days, have students leave the number sentences to be solved by the other half the following day. Of course, students can also do the activity with other students in their class that day.
  • If students are learning from home using Zoom or Google Meet, they can post their number sentences in the chat for other students to solve.
  • Students at home can include their number sentences in homework they return either in packets or electronically and receive number sentences submitted by other students returned in the same format (homework packets, email, etc.)

If students have unreliable Internet access, the Spirit Lake Demo can be downloaded on Mac or Windows computers and played off line. Installers are available for Windows and macOS.

Lesson 6 – Multiplying by 6

Tell students that this week they are going to learn how to multiply numbers from 6 to 9. Now that they know most of their multiplication tables from 0-5, they are ready to play Making Camp at the end of the lesson. Start with this Google slides presentation that introduces multiplication and arrays. Feel free to modify the slides to provide more explanation as needed by your students.

Teaching students at home and need to print out the slides? Click here for a PDF.

End the lesson by playing Making Camp Premium or Spirit Lake for 15-30 minutes.

Lesson 7 – Multiplying by 7

Depending on your students’ progress and interest, you may want to hold off on this lesson for another day or jump right into it after the six tables. Personally, I find it works much better if these lessons are spread out across at least one day each. Today’s lesson starts with a Google Slides presentation that practices multiplying by 7.

Teaching students at home and need to print out the slides? Click here for a PDF.

End the lesson by playing Making Camp Premium or Spirit Lake for 15-30 minutes.

Lesson 8 – Multiplying by 8

Almost there! For a little variety, start the lesson with a game, either Making Camp Premium or Spirit Lake. After playing the games, for even more variety, this Google slides presentation shows students how to insert images in Google doc or slides files so they can create their own illustration of concept like 8 x 6 = 48 .

Teaching students at home and need to print out the slides? Click here for a PDF.

8 x 6 = 48 pigs

Lesson 9 : Multiplying by 9

Hurray! We did it! Today’s lesson covers multiplying by 9, with skip counting, a brief review of arrays, students creating their own number sentences and a little variety with students finding objects around their class or home to show 9 x 9 = 81 or other number sentences you assign/ they create. You may want to modify the Google slides presentation for today to emphasize any area where your students struggled or you think they need a refresher.

9 x 9 = 81 dots

After the presentation, students should play Making Camp for 15-30 minutes. If you would like stickers, pencils or other incentives to give your students to congratulate them for learning their multiplication tables to 100, just email support@7generationgames.com and we’ll be happy to send you a teacher gift pack.

End the lesson by playing Making Camp Premium or Spirit Lake for 15-30 minutes.

Teaching students at home and need to print out the slides? Click here for a PDF.

Assessment

Each lesson includes multiple assessment:

  1. Students complete the multiplication tables – these can be shown with a projector (in class), on a screen, if teaching remotely, and on paper for students learning at home.
  2. Students write their own number sentences using the multiplication learned each lesson.
  3. Students complete the problems written by their classmates.
  4. Optionally, students write word problems.
  5. Optionally, student complete word problems written by their classmates.
  6. Making Camp Premium and Spirit Lake both have teacher reports showing the number of multiplication problems attempted and answered correctly.

State Standards

Minnesota State Standard 3.1.2.3 – Represent multiplication facts by using a variety of approaches, such as repeated addition, equal-sized groups, arrays, area models, equal jumps on a number line and skip counting. Represent division facts by using a variety of approaches, such as repeated subtraction, equal sharing and forming equal groups. Recognize the relationship between multiplication and division.

Minnesota State Standard 3.1.2.4 – Solve real-world and mathematical problems involving multiplication and division, including both “how many in each group” and “how many groups” division problems.

Minnesota Math Standard 3.2.2.2 – Use multiplication and division basic facts to represent a given problem situation using a number sentence. Use number sense and multiplication and division basic facts to find values for the unknowns that make the number sentences true.