Worksheets on multiples and factors: Factors and Multiples Worksheet
Posted onFactors and Multiples Worksheets — Ashleigh’s Education Journey
These factors and multiples worksheets and activities can help you as you teach factors and multiples. I speak from experience when I say that teaching factors and multiples is not for the faint of heart. In fact, factors and multiples are one of the most challenging concepts for students. A lot of that challenge stems from students not knowing their multiplication facts, but we can’t skip that concept because of that lack of core or background knowledge.
Hopefully as we teach factors and multiples, students will develop more efficient strategies for multiplying numbers and will continue to improve their memorization of multiplication facts. In this post, I’ve shared some of the lessons I use from my Multiplication Unit that help students understand factors and multiples.
In this post:
- What Are Multiples
- Multiples Booklet
- What Are Factors
- Factor Game
- Sort
- Prime and Composite Numbers
- Extra Practice
- Additional Resources
What Are Multiples
In the factors and multiples worksheets and lessons, students start by learning about multiples which was fairly easy for most students. During these lessons, a hundreds chart may be students’ best friend. They use hundreds charts to highlight or color multiples of two, three, four, five, and six. During the lesson, have students look at the pattern on their hundreds chart and ask students to predict other numbers that are multiples of that number.
Multiples Booklet
After introduce multiples there are a few different options. Students can complete this multiples booklet where the write the multiples through one hundred for digits 2-10.
An alternative to the booklet is a lesson where students fill in dots for multiples of numbers through 100. It gives students another way to look at number patterns. It’s definitely a unique twist to traditional factors and multiples worksheets.
What Are Factors
Identifying factors is much more difficult for students. In the first activity, students used a stack of large index cards to create a number line from 0-100. I gave each student an index card, and on the index card they had to show all of the factors of that number. I gave students 1 cm and .5 cm grid paper to create arrays to represent their number. For instance, the student who had 12 would have created a 1×12, 2×6, and 3×4 array. Students also wrote the factor pairs for their numbers. Some students quickly recognized that some numbers had several factors, while other numbers (prime numbers) only had one factor pair. After students finished their first index card, they received a new index card and repeated the steps above until we had an index card for every number 1-100.
Once we had all 100 index cards ready, we taped the cards to the wall in the hallway to create a super long number line. Then, students were able to go on a number walk to identify all of the prime numbers through 100. This was such a meaningful activity for my students, and I felt like building the arrays and discovering which numbers were prime and which were composite made a pretty big impact on my students.
Students often need extra time to understand factors and multiples, so you can also incorporate a different version of the lesson. This factor worksheet is a bit more scaffolded and includes grids for students to draw all possible arrays for numbers 1-21.
Prime and Composite Numbers
This is one of my favorite factors and multiples worksheet, because The Sieve of Eratosthenes is my absolute favorite way to teach prime and composite numbers. It’s fascinating for students to see how they can eliminate composite numbers! There is a great corresponding video on BrainPop that I like to incorporate with this lesson.
Factor Game
It’s important to not go too fast and leave students who need extra instruction behind. After I introduced factors, I wanted to take a day to work with students on an as needed-basis, so I introduced a fun factor game. While the majority of my students were playing the game, I met with those students who needed additional instruction. In the game, students roll two dice and use the dice to create a two-digit number. After students create the number and record it on the recording sheet, they write all the factors of that number on the same line. Then, students write the next five multiples of the number. Have students highlight or circle all the prime numbers they created.
Factors and Multiples Sort
I LOVE using a Venn-Diagram for a factors and multiples sort. This is an open sort so, students determine the criteria for each section. Students select two labels for the first sort and three labels for the second sort. I did require students to select criteria that would include at least one number in each section.
Factors and Multiples Extra Practice
If you see that your students need a little extra practice with factors and multiples, you might enjoy some of the extra practice pages that I included in the unit.
While this was a bit challenging, I do think that it required students to think in a way that would allow them to develop a better conceptual understanding of factors and multiples. You can download a free copy of the assignment here.
Additional Resources
You can have students add factors and multiples to their Math Reference Notes (top) or Math Scribble Notes (bottom). Throughout the past two weeks, I have been impressed with how many times my students referred to these notes.
I also incorporate factors and multiples into a number of the day routine. Each day of the school year, introduce a new number. As a class, count by the multiples of that number through 100. Then, find all the factors for that number. After you finish a number, add it to a class banner and display in the classroom for student reference.
I print the prime numbers on teal paper, so they all look the same and really stand out. I print the composite numbers on white paper have students take turns decorating the composite numbers for the banner. This gives students a little more ownership of the banner. You can find the banner here.
I don’t begin this factor of the day until after I introduce this unit, so during the first lesson, we may complete numbers 1-30 (or whatever day of school we are currently on).
To be 100% transparent, I tried eliminating that activity this year, and it was such a mistake! Come Monday morning that will be added back to our routine!
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Technological map of the lesson «Decomposition of the difference of squares into factors»
Stage of the lesson |
Teacher’s activity |
Student activities |
Formed UUD |
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torque. |
Slide No. 1 “Mathematics should be taught after that, that it puts the mind in order.” M.V. Lomonosov “Mathematics is the gymnastics of the mind” AVSuvorov Let’s follow these words: let’s be active, attentive, absorb knowledge with great desire. On the assessment sheet, you will post the marks you received for each step of the lesson. |
Listen to the teacher, psychological attitude. Sign the self-assessment sheets. |
Personal: be aware of your emotions, reasonably evaluate your own and other people’s actions, based on universal moral values Communicative: understand the position of another Regulatory: activity planning |
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Preparing students for active learning. |
Slide no. 6 no. Slide #7 #2 Read the expressions: Slide No. 8. No. 3 Read the expressions: Oral exercises will help us in further work. We will put the grades for oral work on the self-assessment sheet. -What groups can these expressions be divided into? (square of the difference of expressions and the difference of squares of expressions) What more? Determine the topic of the lesson and define the goals |
Read the expressions: — the square of the difference of two expressions; — difference of squares of two expressions; — the product of the difference and the sum of expressions. Give marks for oral work in the self-assessment sheet |
P: analyze, generalize, compare, present information in different forms; proficiency in performing oral, computational skills C: consciously use speech means, understand the position of another, express one’s opinion, correct one’s opinion P: activity planning, put forward versions L: be aware of your emotions, reasonably evaluate your own and other people’s actions, based on universal moral values, show attention and interest in the educational process. |
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3. Studying the material Motivation for learning activities. Slide Let’s think about what the topic will be at the lesson today if we have learned a new topic. And how will we achieve this goal? Purpose Learn to apply the formulas «factoring the difference of squares into factors» when simplifying expressions, when calculating the values of expressions and solving equations. |
Students formulate the purpose and objectives of the lesson: — develop the ability to apply the difference of squares formula for identical transformations of expressions — be able to factorize polynomials; — be able to solve incomplete quadratic equations of the form ax² — b ^{ 2 } = 0; — be able to apply the formula for calculations; — be able to apply a formula to simplify expressions; — be able to formulate a rule; |
L: show attention and interest in the learning process. Subject: analyze, summarize, compare, present information in different forms Regulatory: evaluate the results of educational activities, analyze one’s own work. |
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Math dictation |
Option 1. x ^{ 2 } – 49 a ^{ 2 } – 9 b ^{ 2 } c ^{ 2 } 4x ^{ 2 } – ^{ 2 } 0.09a 90 165 2 — b ^{ 2 } Variant 2.y ^{ 2 } -9 m ^{ 2 } 90 077 — 4 a ^{ 2 } c ^{ 2 } (1/25a ^{ 2 } -4/9b ^{ 2 } ) 0077 -y ^{ 4 } ) Slide 10 Dictation check, self-assessment |
Check on slide No. 10 Assess the dictation on the self-assessment sheet |
Subject: possession of the skills of simplifying numerical and alphabetic expressions. |
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Journey through history. |
Slide 11 Student’s story about ancient Greek mathematician Diophantus (3rd century AD), Slide 12. Student’s story about Swiss mathematician L. Euler (18th century) |
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4. Consolidation of knowledge. Creative application and acquisition of knowledge, development of methods of activity. Exercise for the eyes. |
Slide No. 13 Today you will play the role of «researchers» in order to consolidate the application of the difference of squares formula for identical transformations of expressions.
(e, h) 2nd variant ** i, c V-1 V-2 d) a=0.5; a = — 0.5; e) x=1, x=-1 g) x=3/2 x= -3/2; h) x = 4/5 x = -4/5 MUTUAL CHECK What task did we implement? We will put marks for the work in the notebook on the evaluation sheet. |
#886 010 a ) 36 / (13-11) (13 + 11) \u003d 3/4 b) (79-65) (79 + 65) / 420 \u003d 4.8 No. 890 a) (x-4) (x +4)=0 x _{ 1 } =4, x _{ 2 } = -4 Give marks for written work in the self-assessment sheet |
L: reasonably assess their own and others’ actions, show attention and interest in the learning process, evaluate their own learning activities. P: analyze, summarize, compare, present information in different forms, establish analogies, master semantic reading, master the skills of simplifying numerical and alphabetic expressions . C: consciously use speech means, understand the position of another, express one’s opinion, correct one’s opinion, organization of work in pairs |
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Performing the task «Crypted word» on the cards: Work in pairs (1 option for weak students) Differentiated tasks: 1 group low level 9000 9 Group 2 medium level Group 3 high level Calculate, choose the correct answer and complete the table. Encrypted word — check on slide 14 900 08 (5 b — 2а )( 5 b + 2а ) a 905 56(7y ^{ 2 } – 6s ^{ 3 }) 0011
Version 3 47 ^{ 2 } -37 ^{ 2 9016 6 (e)53 2 -63 2 (l)126 2 – 74 2 (th) } 21.3 ^{ 2 } — 21.2 ^{ 2 } (e) 0.849 ^{ 2 } – 0.151 ^{ 2 } (p) 52 ^{ 2 } -42 ^{ 2 } What task did we implement? Which words are related in meaning and why do you think so? And what word is superfluous? Task: who was Euler? Let’s put marks for the task in the self-assessment sheet. Check yourself Slide № 14-15 |
Complete the task on the card. Check the task on slide 15 Answers: Option 1 x ^{ 2 } — y ^{ 2 } 9m ^{ 2 } — 16n ^{ 2 } 4a ^{ 2 } -9b ^{ 2 } 36a ^{ 2 } -1 25b ^{ 2 } — 4 a ^{ 2 } — spring Variant 2 — vitamin 5 2 -0.09 a ^{ 2 } -144 64 — 9u ^{ 2 } 49y ^{ 4 } -36 c ^{ 6 } 16b ^{ 2 } c ^{ 2 } – 0.25 c ^{ 2 } -m ^{ 2 } Variant 3 4.25; 10400; 1160; 840;0.698 Euler . Students put an assessment in the self-assessment sheet |
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5. Monitoring the results of educational activities. |
Slide № 15 What task did we accomplish? Did you cope with the tasks of the lesson? |
Performing the test in workbooks. Students compare the checked work with the results of the slide, mark the grade for the test in the self-assessment sheet to their desk mate. Each student sums up in his/her own self-assessment sheet. |
L: be aware of your emotions, reasonably evaluate your own and other people’s actions, develop a respectful attitude towards others, master social roles and rules, show attention and interest in the learning process, evaluate your own learning activities. P: analyze, generalize, compare, present information in different forms, mastering the skills of simplifying numerical and alphabetic expressions. C: consciously use speech means, understand the position of another, express one’s opinion, correct one’s opinion, organization of work in pairs 008 Slide #17 Open your diaries, write down your homework: — repeat the difference of squares rule; — complete No. 888 (d, f), No. 892 (1.2 st.) |
Students write homework in the diary. |
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7. Reflection. Summing up the lesson. |
And now guys continue the sentence: Slide number 16 Today at the lesson I learned … I liked it at the lesson today… Today at the lesson I repeated…and consolidated… What types of work caused difficulties and need to be repeated? What knowledge do you have confidence in? Did the lesson help to advance in knowledge, skills in the subject? Who should work on what? Thank you for the lesson! A brief assessment of the activities of students in the lesson. |
The guys continue the sentences and answer the questions. |
L: to be aware of one’s emotions, reasonably evaluate one’s own and other people’s actions, based on universal moral values, develop a respectful attitude towards others, master social roles and rules R: analyze, generalize, compare, establish an analogy consciously use speech means, correct one’s opinion R: evaluate the degree and ways of achieving the goal |