I have 1 penny in one pocket. I have 6 more pennies in another pocket. This is a Join or “some and some more” story structure.
Many teachers I work with have asked for advice on problem solving in math. Students often don’t know how to approach them or know what operation to use. Should teachers help students focus on key words or not? What about the strategies such as CUBES, draw pictures, make a list, guess and check, work backwards, find a pattern?
While all of those strategies definitely have their purpose, I find we often give kids so many steps to follow (underline this, circle this, highlight that, etc.) that they lose sight of what the problem is basically about.
In this post, I will focus on two basic questions (who and what) and a simple graphic organizer that will help students think about (and visualize) the actions in a one-step Join story problem. KG and first grade students can act out these actions using story mats or ten frames. Late first through 5th grade can use a part-part-whole box. There are two FREE items offered.
These are the types of problems I will focus on in the next few posts.
Join (also referred to as SSM – Some and Some More)
Separate (also referred to as SSWA – Some, Some Went Away)
JOIN problems have 3 versions:
a + b = ___ (The result is unknown.)
a + ____ = c (How the story changed is unknown / missing addend.)
____ + b = c (The start is unknown / missing addend.)
They can also be referred to as “Some and Some More” stories (SSM). This means, you have some and you get some more for a total amount. These present themselves as additive stories, but it doesn’t necessarily mean you have to add to solve them. The second and third version above are often referred to as missing addend problems. Continue reading →
This week I will focus on subtraction problem structures. There are two types: separate and compare. I suggest teaching these models separately. Also, some part-part-whole problems can be solved using subtraction. I will refer to the same terms as in addition: start, change, result. You can also use the same materials used with addition problems: part-part-whole templates, bar models, ten frames, two-color counters, number lines, and connecting cubes.
The goal is for students to see that subtraction has different models (separate vs. comparison) and an inverse relationship with addition — we can compose as well as decompose those numbers. Knowledge of number bonds will support the addition / subtraction relationship. Here is the same freebie I offered last week you can download for your math files: Addition and Subtraction Story Structure Information The six color anchor charts shown below are also attached here free for your use: Subtraction structure anchor charts
Separate: Result Unknown
Example: 10 – 4 = ____; There were 10 cookies on the plate. Dad ate 4 of them. How many are left on the plate?
Explanation: The problem starts with 10. It changes when 4 of the cookies are eaten. The result in this problem is the answer to the question (how many are left on the plate).
Teaching and practice suggestions:
Ask questions such as: Do we know the start? (Yes, it is 10.) Do we know what changed? (Yes, 4 cookies were eaten so we take those away.) How many cookies are left on the plate now? (Result is 6.)
Reinforce the number bonds of 10: What goes with 4 to make 10? (6)
Draw a picture to show the starting amount. Cross out the items to symbolize removal.
Show the problem in this order also: ____ = 10 – 4. Remember the equal sign means the same as — what is on the left matches the amount on the right of the equal sign.
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
Six is 5 and 1. Six is 3 and 3. Or the child may see the bottom as 4 and 2.
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.
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 →