CCSS.MATH.CONTENT.4.NF.A.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2
CCSS.MATH.CONTENT.5.NF.A.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem.
Technology Required
Students will need a Mac or Windows computer or an iPad to play the Fish Lake game. Alternatively, students can play Forgotten Trail on the web using Chromebooks or any computer with a web browser.
Time
35-45 minutes
Lesson Summary
Students play 2-3 levels of a game that teaches and assesses adding and comparing fractions with different numerators and denominators, with the context of a story from Ojibwe history. They create their own problems using visual models to compare fractions. Students discuss classmates’ problems. The lesson culminates with a video on visualization as a problem-solving strategy.
Lesson
Start with a Game (10-20 minutes)
Students play the Fish Lake game through level 2. This requires installing the game on an iPad or a Mac or Windows computer. If only Chromebooks are available, students can play the Forgotten Trail Game instead. We do recommend Fish Lake if iPads are available. Although mathematics and social studies standards taught in both games overlap, the change from using Chromebooks in most lessons to a more console-level game can improve student engagement.
Students create their own problems (10 minutes)
Use slides 1-4 of this Slides presentation to explain the assignment. Optionally, for students learning at home or for homework, it can be assigned to students in Google classroom or similar system.
Discuss math problems students created (10 minutes)
Because students very often ask, “What do you mean?” or “Give me an example?” slides 5-11 of the presentation give an example.
A visual model of equivalent fractions
Watch a video (2.5 minutes)
Visualize! One way of solving a problem
Assessment
This lesson includes three types of assessment. In the games, students are presented with math word problems that relate to the game narrative. Their answers are scored and data can be accessed for each student from the teacher reports. Students submit math problems they have written for teacher feedback. Also, teacher can use whole class discussion of student problems as a gauge for understanding or have students in pairs or small groups submit written discussion of their peers’ math problems.
Students are introduced to the various means of problem solving in a brief presentation. They watch a video on visualization, then solve a problem that asks them to visualize. After watching a video on building a model, students build and/or draw their own model of a multiplication problem or property. Lesson concludes with game play to reinforce these problem-solving strategies and learn more. This lesson assumes that students have some familiarity with multiplication of one-digit numbers and division with one-digit divisors.
📲 Technology Required
If teaching in person, the teacher will need a computer and projector or smart board to show the videos, or students can be given the links to watch on their own devices. Students will need a PC, Mac or Chromebook or tablet. Making Camp Bilingual, Making Camp Lakota and Making Dakota are all playable on any web browser on those devices. Spirit Lake: The Game, playable on Mac or Windows computers also teaches these same concepts. Students will also need access to the games.
📚Lesson
Watch a Video: Visualize (2:38)
Or watch it in Spanish – Visualizar (2:38)
This video teaches you how to visualize a problem. Visualization may help you solve it more easily. If you use objects to imagine quantities of numbers in certain problems, you will be able to find the solution faster.
Short presentation and problem (10 minutes)
After watching the video, teachers can present this short Google slides presentation or assign to students in Google classroom, giving another example of visualizing a problem.
You can simply save to your own Google classroom and then assign to your class to include both types of problems or modify to add more explanation or delete whichever one does not apply to your class before assigning.
Play a game (15 minutes)
Assign students to play Making Camp Bilingual (plays on any device) to practice these problem-solving strategies and more with multiplication and division.
Two assessments are built into the lesson, on visualization and building a model. Teachers can also see which standards students have attempted and how many problems they have answered correctly in the Making Camp Bilingual teacher reports.
The lesson above has a companion lesson for English only learners. Problem-solving Two Ways is the same lesson from above but provides the resources, videos and Google Slides, in English only.
CCSS.MATH.CONTENT.3.OA.D.8 Solve two-step word problems using the four operations. 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.
⏰ LESSON TIME:
30-45 minutes
📃 SUMMARY
In this lesson, the students will develop an understanding of the meanings of the four operations of whole numbers through activities and problems involving real life scenarios from Indigenous history. Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve using the four operations problems involving single-digit factors. It includes educational videos, games and video presentations that can be used for reviews and daily practice.
📲 TECHNOLOGY REQUIRED
Computer or mobile device
📚 LESSON
First, watch a video for 100 second
Learn how to solve problems by breaking them down into easier problems (time 1:40)
Second, play a game
Choices screen: Instruct students to select NUMBERS
Let the students log in to Making Camp Premium to access the game. (Game links can be found under the GAMES tab.) The choices screen is shown above. Students are instructed to select NUMBERS from the screen above.
Students are tasked to play for 20 minutes. Students will demonstrate mastery in solving word problems like the one shown below and to be able to earn points within the game. For some students who aren’t able to master the standard will be given individualized instruction to master the concept.
Example of word problem following a video
Reinforce and assess what they have learned with a presentation
This Google slides presentation, Travois and Multiplication, gives a little background on how the travois was used by Indigenous people of North American. Within the context of building and using a travois, students learn to analyze a word problem and use appropriate operations in getting the correct answer.
Final video – See travois in action
Watch this short (46 second) video on how Lakotas used the travois
ASSESSMENT
To monitor the progress of the students, there is data analysis that can be viewed on the Making Camp Premium data report. There are also four problems included in the Google slides presentation that students can complete in-class or as a work-from-home activity
This cross-curricular lesson includes activities and instruction in agricultural science and math. Students begin by watching a video and learning about pig farms. After making their own pig barn, they watch two short videos about solving math problems. This information is then applied to solve problems during a presentation on math around the pig farm. Students end playing one of the Making Camp games to reinforce skills and knowledge.
📲 Technology Required
If teaching in person, the teacher will need a computer and projector or smart board to show the videos, or students can be given the links to watch on their own devices. Students will need a PC, Mac or Chromebook or tablet. Making Camp Premium, Making Camp Lakota and Making Dakota are all playable on any web browser on those devices.
📚 Lesson
This lesson starts with resources from National Ag in the Classroom
Virtual tour of pig farms
Virtual Field Trip to Ohio Pig Farms
Make a Pig Barn
This activity requires a few supplies but it is probably things you have lying around and your students will probably enjoy it.
Business-size envelopes, 4 per group
Paper towel, 1 per group
Scissors
Scotch tape
OPTIONAL
Markers
Toilet paper rolls, 2 per group
Drinking straws, 2 per group (cut into 8 equal pieces)
8.5″ x 11″ white paper, 1 per group (cut in half)
Extra paper for making fencing, pipes, feed troughs, etc. (optional)
Instructions
Use the following instructions to model for the students how to create the barn:
Barn: Cut an oval hole in one envelope, making a large side window for the barn. This window provides the proper ventilation for the pigs.
Cut the paper towel in half and tape it onto the top of the window for the curtain.
Cut another envelope in half for the ends of the barn.
Tape the ends of the barn to the “sides of the barn” envelopes, one of which has the hole for the window and paper towel curtain, so that you have four sides, or a rectangle.
Use the final envelope to create a roof by creasing it in half lengthwise and attaching it with tape to the top of the rectangle.
Food Storage: Tape four straws, or legs, to each toilet paper roll so that the structures will stand on the legs.
Use a half piece of paper, and make a cone shape by twisting and taping the ends. Tape the cone shape on the end of the toilet paper roll without the straw legs.
Use the other half piece of paper to make another smaller cone shape and tape it between the straw legs on the other end of the toilet paper roll.
If you’d like, you or your students can watch the instructions here. You can also assign this video for students learning at home to watch so they know how to make the barn. The plus is that just about every house will have every single one of these items except possibly the straws.
Optional additional science and language arts content
This link to the National Ag in the Classroom lesson has more information on pigs and pig farming, including some of the vocabulary used in the math lesson as well as a discussion of the ways farmers care for animals. I highly recommend checking it out.
Watch a video on operations key words
Trust me, this does come back to pig farming!
Students watch this video on operations keywords. This two-minute video has been watched over 14,000 times, which gives some indication of how useful students and teacher find the concept of looking at the words in a problem to decide which operation to use.
Watch a video on problem-solving – Start at the end
This 3 1/2 minute video explains that the end of the word problem is where you usually will find the question you are expected to solve. It includes one easy and one harder example, as well as a couple of useful tips.
Start at the End
Give a presentation
This 34-slide deck on problem-solving reinforces the information in the two math videos and gives students three problems of increasing difficulty where they have to start at the end, all centered around Laura’s pig farm.
Play a game
Now that students have been introduced to problem-solving with multiplication and division word problems, it’s time to play a game and reinforce those skills. Which game depends on what you feel your students need most. There is overlap among the games as each includes some review.
Making Camp Premium – focuses primarily on multiplication of one- and two-digit numbers. Also includes division with one-digit divisors. The content is taught in the context of Ojibwe history and culture.
Making Camp Lakota – focuses primarily on division with one-digit divisors. Also includes multiplication of one- and two-digit numbers. The content is taught in the context of Lakota history and culture.
Making Camp Dakota – focuses primarily on division of three-digit numbers with one- and two-digit divisors. Also includes multiplication of one- and two-digit numbers and division with one-digit divisors. The content is taught in the context of Lakota history and culture.
Assessment
Assessments are built into the presentation, as teachers can have students submit their answers in writing or in a Google chat prior to giving the answers during the presentation. Teachers can also see which standards students have attempted and how many problems they have answered correctly in the Making Camp teacher reports.
As the title suggests, this lesson introduces students to two other problme-solving strategies. They watch a video on visualization, then solve a problem that asks them to visualize. After watching a video on building a model, students build and/or draw their own model of a multiplication problem or property.
CCSS.MATH.PRACTICE.MP4 Model with mathematics. CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them. CCSS.Math.Content.3.OA.B.5 Apply properties of operations as strategies to multiply and divide.
⏰ Time
30-40 minutes
📃 Summary
Students are introduced to the various means of problem solving in a brief presentation. They watch a video on visualization, then solve a problem that asks them to visualize. After watching a video on building a model, students build and/or draw their own model of a multiplication problem or property. Lesson concludes with game play to reinforce these problem-solving strategies and learn more. This lesson assumes that students have some familiarity with multiplication of one-digit numbers and division with one-digit divisors.
📲Technology Required
If teaching in person, the teacher will need a computer and projector or smart board to show the videos, or students can be given the links to watch on their own devices. Students will need a PC, Mac or Chromebook or tablet. Making Camp Premium, Making Camp Lakota and Making Dakota are all playable on any web browser on those devices. Spirit Lake: The Game, playable on Mac or Windows computers also teaches these same concepts. Students will also need access to the games.
📚 Lesson
Watch a Video: Visualize (2:38)
This video teaches you how to visualize a problem. Visualization may help you solve it more easily. If you use objects to imagine quantities of numbers in certain problems, you will be able to find the solution faster.
Short presentation and problem (10 minutes)
After watching the video, teachers can present this short Google slides presentation or assign to students in Google classroom, giving another example of visualizing a problem. The presentation ends with asking students to visualize another problem themselves. It then leads into the next video.
Have students create a model using Google draw, jam board or simply drawing it on paper.
Build a physical model using objects in their home or school
Google slides presentation describing these assignments is given here. You can simply save to your own Google classroom and then assign to your class to include both types of problems or modify to add more explanation or delete whichever one does not apply to your class before assigning.
Play a game (15 minutes)
Assign students to play Making Camp Premium (plays on any device) or Spirit Lake (plays on Mac and Windows computers) to practice these problem-solving strategies and more with multiplication and division. Play Making Camp Lakota to reinforce division skills and concepts.
Two assessments are built into the lesson, on visualization and building a model. Teachers can also see which standards students have attempted and how many problems they have answered correctly in the Making Camp teacher reports.
The lesson above has a companion lesson for English Learners. Problem-solving Two Ways (Bilingual English & Spanish) is the same lesson from above but provides the resources, videos and Google Slides, in English and Spanish.
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.
⏰Time
35- 45 minutes
📲Technology Required
Computer with a projector, Smartboard or other device for your class to watch videos
NOTE: You’ll also need 20 math problems: 3-digit numbers multiplied by a 2-digit number. You can use what you already have, use the worksheet provided or use the Multiplication Worksheet Generator to make your own worksheet.
📃Lesson Summary
Students practice multiplication for a few problems, watch a video on using estimation to solve problems, hear a brief presentation from their teacher, solve more practice problems, watch a second video and practice more problems. They end the lesson with a game and a discussion of which strategies proved most useful.
📚Lesson Plan
Start with multiplication problems
Have the students solve 3 to 5 problems.
NOTE: This lesson requires a total of 9 to 15 multiplication problems. This worksheet has 24 problems multiplying a three-digit numbers by a two-digit number. Teachers can assign any selection of these problems or use their own. We give the students a worksheet of 24 problems and tell them to pick any ones they want to solve. Of course, teachers should do whatever works for them.
Practice multiplying 2-digit and 3-digit numbers again
Now that students have had a chance to watch and discuss the video showing long multiplication, step by step, it’s time for them to practice again. They have 10 minutes to complete another 3 to 5 math problems.
Watch another video to review the steps
Now that students have watched one video and practiced their skills with a few problems they watch a second video to reinforce those steps.
Practice multiplying 2-digit and 3-digit numbers again
Now that students have had a chance to watch and discuss the video showing long multiplication, step by step, it’s time for them to practice again. They have 10 minutes to complete another 3 to 5 math problems.
Peer-grading (optional)
You may wish to share the answer key with students and have them grade their own or their peer’s problems.
Discussion
As a class, discuss strategies for solving problems, including doing a reality check, estimation and breaking problems into smaller problems. Ask students which strategies worked best for them.
Play a Game
Both Spirit Lake: The Game and Making Camp Dakota (released in February, 2021) teach multiplication of two- and three-digit numbers.
CCSS.MATH.CONTENT.3.OA.D.8 Solve two-step word problems using the four operations. 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.
In this lesson plan, students will learn about estimation as a problem solving strategy and practice rounding to the nearest 10 and 100. Including 0. The lesson will reinforce rounding up for numbers 5 and greater. Through gameplay and an interactive assessment activity that includes multi-step problem solving, students will practice rounding with an emphasis toward developing mastery.
📲 TECHNOLOGY REQUIRED
Device with web-browser (Chromebook, laptop or desktop computer); or iOS (iPhone/iPad) with access to Google apps.
📚 Lesson Plan
Watch the video– Problem Solving: Estimate (optional if playing Making Camp)
This animated movie explains how estimation and rounding can be used to solve problems that don’t require exact calculations, including multi-step word problems with an example. It also explains how rounding can be used to make numbers easier to work with and includes an example with rounding to the nearest hundreds. (3:44)
2. GAME: Making Camp Premium
Have students play Making Camp for 15-20 minutes, including this icon activity on estimation.
This activity will also reinforce visual estimation skills.
ASSESSMENT
A Day at the State Fair: Estimation is Everywhere
If your students have not previously used virtual manipulatives you may want to watch this short video.
You may elect to read and work through it during class time, giving students time to complete the problems or assign it as an individual assignment. Estimated time to complete: 10-15 minutes.
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.
Minnesota Math Standard 2.1.1.4 – Round numbers up to the nearest 10 and 100 and round numbers down to the nearest 10 and 100.
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.
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. (Common Core Standards)
H.3-5.3 Describe the North Dakota Native American Essential Understandings. (US State – ND – Social Studies 2019 Standards)
📃 Summary
In this integrated lesson, students learn about responsibilities of children in traditional Dakota society and discuss their responsibilities today. They learn a problem-solving strategy that can be applied to a wide range of situations, including mathematics. Students play Spirit Lake: The Game or watch videos solving multiplication problems set in the context of a story based on Dakota culture. Students can complete a written assessment in class and on-line classes can complete these problems in Google doc or other preferred method .
📲 Technology required
The game in this lesson plan, Spirit Lake, can be played on Mac or Windows computers. Once students have logged in, they can continue playing, even if the Internet drops. The program should be installed on devices before the lesson. Students will create a username and password the first time they play the game. If students need to play offline, they can play Spirit Lake Beginnings – Lakota . If students do not have access to a Mac or Windows computer they can watch similar content in the linked videos.
⏰ Time Required
30-40 minutes
📚 Lesson Plan
1. Presentation on responsibilities and problem-solving
Begin with the PowerPoint presentation discussing student responsibilities. Also available as Google slides. Discuss with students the responsibilities they have at home. Explain that whether students are baking a cake or solving a multiplication problem they can take the same four steps to solve their problems. You’ll also use this presentation for practice during the lesson and assessment at the end of the lesson.
Time: 5-10 minutes depending on length of student discussion.
Play the Spirit Lake game, In levels 1 and 2 of the game, players solve multiplication problems set in the context of a story based on the history and culture of the Dakota people. They learn a problem-solving strategy that can be applied to a wide range of situations, including mathematics. .After this lesson, students will be able to solve word problems using multiplication of one-digit numbers and understand the general application of problem-solving steps. (10-15 minutes)
Option for students without access to Mac/ Windows computers
3a. Game Play Video
For schools that want the game play and multiplication problems from Spirit Lake: The Game but now your students are at home, without a computer, what do you do? You can have them watch this four-minute preview up to the multiplication word problem. Your students will learn a few words in Dakota, see the game and be set for your lesson on multiplying one-digit numbers.
3b Instructional Video
In this six-minute video, students learn four steps in solving a problem: understand what’s required, select a strategy, do it, and test it. Multiplication word problems are used as examples, but these scaffolding strategies for Grades 3-5 can be applied to any problem in math.
Assessment
Individual Assessment
Have students work independently to solve the problem in the PowerPoint or Google slides linked above. Have students present their solutions. Show one recommended solution that is in the presentation.
If students will be completing the assignment in class, print out pages 11,12 and 14 for assessment. On-line classes can complete these problems in Google doc or other method used by the teacher for assignments.
Estimated time: 5 minutes
A second assessment is in the reports for the Spirit Lake game, which show whether students answered correctly the problems in Levels 1 and 2, 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. Note that these reports are NOT available for Spirit Lake Beginnings, which is played offline.
Group Assessment
Continue with the PowerPoint presentation or Google slides. Students will work in groups to solve 4 multiplication problems. Estimated time: 5 minutes
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.
