This is part 4 in a continuing series of posts about basic multiplication teaching concepts. Use them for beginning lessons or reteaching for struggling learners. Students could be struggling because they were not given enough exposure to concrete and pictorial models before going to the numbers only practices. The focus in this post will be skip counting to determine multiplication products. I will even focus on skip counting done in early grades (counting by 10’s, 5’s, and 2’s). Read on for 10 teaching strategies regarding skip counting.
I am going to give some of my opinions and misconceptions students have about skip counting.
Many students do not associate skip counting with multiplication, but just an exercise they started learning in KG and 1st (skip counting orally by 10’s, 5’s, and 2’s). This is often because they started with numbers only and did not have the chance to see what this looks like using concrete objects or pictorial representations.
If you observe students skip counting, are they really just counting by 1’s over and over again? Or are they adding the number they are skip counting by repeatedly. You know the scenario. You tell a student to skip count by 3’s and they know 3, 6, 9, but then hold up their 3 fingers and count 10, 11, 12, 13, 14, 15, 16, 17, 18, and so on. Or are they truly counting like this: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30?
The main issue I have with skip counting is that if a student makes an error regarding just one of the numbers in the sequence, then the rest of the sequence is incorrect. So this should not be their only strategy. Do you recall a previous story I mentioned about the 5th grader who tried to solve 12 x 3 by skip counting on a timed facts test? He was unsuccessful because he kept losing track and didn’t have another strategy to use.
Successful skip counting reinforces the concept that multiplication is repeated addition – do your students know this? I have witnessed many students who know the first 2-3 numbers in a skip counting sequence, but then don’t know how to get to the next numbers in the sequence.
Students don’t often relate the commutative property to skip counting. Let’s say the problem is 5 x 8. The student tries skip counting by 8’s (because this problem means 5 groups of 8) and may have difficulty. Does the student try to skip count by 5’s eight times instead?
Ten teaching strategies for skip counting:
For young students skip counting, use objects to show how to keep track:
Base 10 rods
Rekenrek (easily slide 5 or 10 beads at a time)
5, 10, 15, 20
Count by 5’s or 10’s
Hand prints (for counting 5’s or 10’s): Which do you think would give students a better understanding: Holding up one hand at a time and counting by 5’s or lining up several children and having them hold up their hands as you continue counting? The second scenario enables students to see the total of fingers as opposed to just 5 at a time.
Use money: nickels and dimes
Associate counting by 2’s with concepts of even and odd
Use manipulatives. Do it often and with a variety of materials. The arrangements should emphasize the other strategies (equal groups, arrays, repeated addition).
Draw and label pictures. The labels for this strategy would show the cumulative totals instead of just the number in each group.
Arrange students in line or groups to practice skip counting. Example if practicing 4’s: Every 4th student turns sideways, every 4th student holds up their hands, every 4th student sits down. every 4th student holds a card with the number representing their value in the counting sequence, etc.
Practice skip counting while bouncing or dribbling a ball. Great for PE class!
Associate skip counting with sports: 2 and 3 pointers in basketball, 6 points for touchdowns in football, etc.
Use a 0-100 chart to see patterns made when skip counting. I love the 0-100 pocket chart and translucent inserts that allow you to model this whole group. Individual 100 charts are readily available in which students can mark or color the spaces. Here are links to the chart and the translucent inserts: 1-100 pocket chart and Translucent pocket chart inserts
Look for other patterns regarding skip counting. Refer to my previous post on this for more details: Skip counting patterns
Skip counting by 2, 3, 4
Skip counting by 6, 8, 9
Relate skip counting to function charts and algebraic patterns using growing patterns.
Practice skip counting using money: by 5’s, 10’s, 25’s, 50’s
What strategies do you like for multiplication? What misconceptions do you see with your students?
Next post will be part 5 of my multiplication posts – and the last one for this school year. I will focus on using these basic concepts with double-digit problems. Stay tuned!!
For my Lawton, OK friends who are implementing Saxon Math but don’t have wall space for all of the math meeting components, here are a few links from TPT for grades KG-3rd for ppt / SMARTBOARD versions. The important aspect of the morning meeting is the interactive part where individual students have a role in placing the numbers, days, coins, patterns, clock hands, graph piece, etc. on the board each day. With that said, there is also value in having some of the math meeting components visible throughout the day such as the calendar, 100 chart, and number of the day. Think of the components you or your students would most likely refer to outside of the math meeting time to keep a permanent physical display.
Here are the links. Read the other purchasers’ comments and look at the previews to get more info. I have not purchased any of these so I can not vouch for the quality or usefulness. For those of you who already have a math meeting ppt that you recommend, please let us know!! Thanks!
In Part 1 I focused on a numerical fluency continuum, which defines the stages a child goes through to achieve number sense. After a child has a firm grasp of one-to-one correspondence, can count on, and understands concepts of more and less, he/she is ready to explore part-part-whole relationships which lead to the operations of addition and subtraction. That will be the focus of this post. Read on for free number bond activities and a free number bond assessment!
One way to explore part-part-whole relationships is through various number bonds experiences. Number Bonds are pairs of numbers that combine to total the target or focus number. When students learn number bonds they are applying the commutative, identity, and zero properties. Do you notice from the chart below that there are 4 number bonds for the number 3; 5 number bonds for the number 4; 6 number bonds for the number 5, etc? And . . . half of the number bonds are actually just the commutative property in action, so there really aren’t as many combinations for each number to learn after all.
KG students should master number bonds to 5.
First graders should master number bonds to 10.
Second graders should master number bonds to 20.Teaching Methods for Number Bonds
Ideally, students should focus on the bonds for one number at a time, until mastery is achieved. In other words, if working on the number bond of 3, they would learn 0 and 3, 3 and 0, 1 and 2, 2 and 1 before trying to learn number bonds of 4. See the end of this post for assessment ideas.
Ten Frame cards: Use counters to show different ways to make the focus number. (See above example of 2 ways to show 6.) Shake and Spill games are also great for this: Using 2-color counters, shake and spill the number of counters matching your focus number. See how many spilled out red and how many spilled out yellow. Record results on a blank ten-frame template. Repeat 10 times.
Number Bond Bracelets: Use beads and chenille stems to form bracelets for each number 2-10. Slide beads apart to see different ways to make the focus number.
Reckenreck: Slide beads on the frame to show different combinations.
Part-Part-Whole Graphic Organizers: Here are two templates I like. Start with objects matching the focus number in the “whole” section. Then move “part” of them to one section and the rest to the other section. Rearrange to find different bonds for the same focus number. Start students with manipulalatives before moving to numbers. Or use numbers as a way for students to record their findings.
7 is the “focus number”
1 and 6 are bonds of 7
Sample recording page
Once students have a good concept of number bonds, these part-part-whole organizers are very helpful when doing addition and subtraction problems (including story problems) using these structures: Result Unknown, Change Unknown, and Start Unknown. Children should use manipulatives at first to “figure out” the story.
Here is an example of a change unknown story: “I have 5 pennies in one pocket and some more in my other pocket. I have 7 pennies all together. How many pennies in my other pocket?” To do this, put 5 counters in one “part” section. Count on from 5 to 7 by placing more counters in the second “part” section (2). Then move them all to the whole section to check that there are 7 all together. Students are determining “What goes with 5 to make 7?” 5 + ___ = 7
Here is an example of a result unknown subtraction story: “Mom put 7 cookies on a plate. I ate 2 of them. How many cookies are still on the plate?” To do this start with the whole amount (7) in the large section. Then move the 2 that were eaten to a “part” section. Count how many are remaining in the “whole section” to find out how many are still on the plate? 7 – 2 = ____.
How are number bonds related to fact families? A fact family is one number bond shown with 2 addition and 2 subtraction statements. Ex: With number bonds 3 and 4 for the number 7, you can make 4 problems: 3 + 4 = 7; 4 + 3 = 7; 7-3 = 4; and 7-4=3.
To be able to add and subtract, students normally pass through several phases as they build readiness for these operations with numbers. As teachers, we know oral counting does not necessarily indicate an understanding of numbers and sets, just like reciting the alphabet doesn’t necessarily mean a child can recognize letters and sounds. Read ahead for freebies in the Part-Part-Whole section.
Numerical Fluency Continuum: There are 7 steps to numerical fluency. If a child gets stuck on any of these steps, it may very likely halt their progress. Hopefully children move through these by the end of 2nd grade, but many students beyond that level have a breakdown which is likely because they missed one of these stages. Can you determine which of these stages your students are in?
One-to-one correspondence: The ability to count objects so each object counted is matched with one number word.
Inclusion of set: Does a child realize that the last number counted names the number of objects in the set? A child counts 5 objects. When you ask how many, can they state “5.” If you mix them up after they just counted them, do they realize there are still 5?
Counting on: If a child counts 5 objects and the teacher then puts 2 more objects for the child to count, do they start all over or continue counting from 5? 5 . . . 6, 7.
Subitizing:Recognize an amount without physically counting (ie on dice, dot cards, fingers).
More Than / Less Than / Equal To: Can a child look at two sets of objects and tell whether the second set is more, less, or equal to the first set. Can a child build a second set with one more, one less, or equal to the first set?
Part / Part / Whole: Compose and decompose sets by looking at the whole and the parts that make up the whole.
7 is the “focus number”
1 and 6 are bonds of 7
Unitizing: The child is able to move from counting by ones to count by sets / groups: fives, tens, etc.
In the next few posts, I will show various ways to conduct daily math meetings which you can incorporate into your daily schedule (as part of your normal morning meeting routine, or at the beginning of your daily math lesson). Daily Math Meetings (10-15 minutes) are vital for quickly reviewing math concepts and number sense in more visual and discussion based format. With primary students, this math meeting might center around the calendar bulletin board (or SMARTboard presentation). With intermediate students, it begins to take on the aspects of a “Number Talk” with a variety of computational strategies being the focus.
PreK – KG Level Components:
Days of the Week
Months of the Year
Graphing (weather, etc.)
Place Value (tens and ones: ten frames, straws, sticks, etc. to keep track of the days of school – working toward the 100th day)
1st – 2nd Grade Level Components:
The above plus . . .
Number Bonds (How can we break apart this number? Such as 10 = 3 + 7 or 6 + 4)
Place Value and skip counting using a 100 chart
Number of the Day (word form, base ten form, place on a numberline, tally marks, on a ten frame, expanded form, etc.)
Ordinal Numbers (using the calendar)
Counting money (add one cent each day and exchange pennies for nickels, nickels for dimes, etc.)
This is such an important process in the continuum of counting, adding, and subtracting numbers. It means students can recognize certain quantities without physically counting each one. Continue reading →
A Number Talk is an opportunity to review number sense and operations by making it part of your daily math routine — so that what has previously been taught is not easily forgotten.
In this post I will expand on 2 methods for conducting a Number Talk session for KG-1st grade students (Subitizing and Number Bonds). Refer to a previous post (Sept. 10 – Daily Practice to Build Number Sense), in which I mentioned several other ways to review math concepts on a daily basis such as calendar topics, weather graphs, counting # of days of school, using a 100 chart, Choose 3 Ways, etc. Continue reading →
Number Bonds are pairs of numbers that combine to total the target or focus number. When students learn number bonds they are applying the commutative, identity, and zero properties. PLUS, the information can be applied to both addition and subtraction problems. Number bonds of 10 are very critical to our place value system, and will enhance a student’s success with future addition and subtraction strategies such as use of an open number line.
To build number sense, students need frequent exposure or review of concepts you have previously introduced. There are many ways to build number sense on an on-going, informal basis – especially when you can squeeze in 10-15 minutes daily:
by C. Elkins, OK Math and Reading Lady (updated post on 8-12-17)
The term “subitize” means to recognize quantity without counting. It is a concept recently added to the new OAS (Oklahoma Academic Standards). KG students should be able to “recognize without counting the quantity of a small group of objects in organized and random arrangements up to 10.” For first graders, the quantity is increased to 20 of “structured arrangements.” Subitizing is an important pre-requisite skill to learning addition and subtraction number combinations or number bonds.
Suggested items for the teacher to present this concept: