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COVID-19 Update: We hope that you are finding ways to keep you and your loved ones safe and healthy. We are being inundated with emails from districts, schools, teachers and parents who are looking for offline activities due to the fact that many families do not have the number of electronic devices needed for multiple children to do online learning while parents work from home. In response to this need we are compiling packets of materials for K-5th Grade which will be available for page. We are also offering a free download on the Subscriber Freebies 20% discount on all items in our eBook Store purchased online through April 20th. Use code Learnathome at checkout.

Using Ten Frames and Dot Cards
Enlarged dot cards can be used during mental math sessions to prompt mental images of numbers and different mental strategies for manipulating these numbers. Activities with enlarged ten frames enable children to automatically think of numbers less than ten in terms of their relationship to ten, and to build a sound knowledge of the basic addition and subtraction facts for ten, which are an integral part of mental calculation. Listed below are a few examples of the many ways that these materials can be used during mental math sessions and/or in math centers to build number sense:

Flash a dot or ten frame card briefly and have students write the number on a whiteboard. Using dry erase boards rather than having students say the number, ensures that all children attempt to respond and allows the teacher to assess class progress. When the response is oral, not all student responses are audible. Encourage students to share the different strategies used to find the total number of dots for cards, "How did you see it?" This can be varied by asking students to write the number and draw the pattern they saw, or by having them build the number flashed on their own blank frame.

Flash: One More
Once students are familiar with the basic patterns, and know them automatically, flash a 10 frame or dot card and ask them to name the number that is one more than the number flashed.
Variation: ask students to give the number that is two more/one less/double/ten more than the number flashed

I Wish I Had 10
Flash a dot card or ten frame showing 9 or less and say, "I wish I had 10". Students respond with the part that is needed to make ten. The game can focus on a single whole, or the "wish I had" number can change each time.
Variation: teacher flashes card and students write the complement of ten on individual whiteboards with dry erase markers.

I Wish I Had 12
As above but students respond with how many more are needed to make twelve. Students should be confident in facts of 10 before this is attempted. For example to go from 8 to 12, they should realize they need 2 more to get to 10, then 2 more to 12. 2 and 2 is 4.
Variation: students draw an empty number line on their whiteboards to show the two jumps used to get to the target number.

1 more/1 less/10 more/10 less
The following four prompts are written on the board:
one more
one less
ten more
ten less
The teacher flashes a dot or 10 frame card as the 'starting number'. The first student selects one prompt. For example, if the teacher flashes a card showing '5' the first student might say, "one more than 5 is 6", the second student might say, "ten more than 6 is 16", and the third student might say, "one less than 16 is 15". Continue until all students have had a turn.

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COVID-19 Update: We hope that you are finding ways to keep you and your loved ones safe and healthy. We are being inundated with emails from districts, schools, teachers and parents who are looking for offline activities due to the fact that many families do not have the number of electronic devices needed for multiple children to do online learning while parents work from home.

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Part-Part-Whole Relationships

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Math Teaching Resources for K-5 Classrooms

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Math Teaching Resources for K-5 Classrooms

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K-5 Math Teaching Resources is a leading New York City based math consultancy firm. Our mission is to transform the teaching and learning of mathematics, improving outcomes for all students in elementary and middle school. Click on a tab below for information on our research-based professional development and customizable consulting services. Our results-driven elementary and middle school consultants are committed to working side-by-side with administrators, coaches and teachers to build exceptional quality math instruction and curriculum for schools and school districts, transforming students' math education and achievement.

On this website you will find mathematics to support standards-aligned math instruction. Use the grade level tabs at the top of this page to download sample resources for each grade level from Kindergarten through 5th Grade or visit our EBook Store to purchase a complete set.

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Research based resources and professional learning opportunities to transform math teaching and learning communities.

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Ten Friends | Peg + Cat | Mathematics, Preschool | Video | PBS LearningMedia

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Ten Friends | Peg + Cat | Mathematics, Preschool | Video | PBS LearningMedia

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Use this video to help children count numbers between 1 and 9 to make 10. Ten friends sing about sticking together. Utiliza este vídeo para ayudar a niños a contar números entre 1 y 9 para hacer 10. Diez amigos cantan sobre quedarse juntos.

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Handout: Counting Collections Template (English)

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Handout: Counting Collections Template (English)

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What are the Characteristics of a Good Math Journal Question?

Tasks that ask students to apply memorized procedures to routine problems require one kind of thinking. Non-routine, or open-ended tasks that ask students to engage with concepts and to make connections between ideas and representations require a very different kind of thinking. Although applying memorized procedures is important for developing fluency and should not be dismissed, engaging in thinking and reasoning at a higher level is necessary to develop problem solving skills. While routine, or closed tasks, may be used to assess students' comprehension of number facts, strategies for solving algorithms accurately, or knowledge of other math facts, open-ended tasks involve investigative responses that stimulate mathematical reasoning and foster the understanding that mathematics is a creative endeavor. Some open-ended tasks may have more than one correct answer but a maximum of correct answers while others may have an infinite number of correct responses. For example the task, There are 16 apples to be put into bowls. Each bowl must have the same number of apples. How many different ways can the apples be put into bowls?' has several correct answers while the task 'I solved a division number story and got a quotient of 7. What might the number story be?' provides almost endless possibilities of acceptable responses. Other non-routine tasks such as 'Would you rather eat 1/3 or 2/8 of a plate of broccoli? Use math to justify your reasoning.' have no one correct answer but instead focus on the student choosing an option and justifying their choice using math reasoning. The important thing to consider when selecting, or writing, a math journal task is whether you are providing opportunities for students to regularly engage in a variety of routine and non-routine tasks in order to develop both conceptual understanding and procedural fluency.

A good math journal question ....

builds in differentiation by allowing for multiple entry points and recording techniques, thereby allowing all students to work at their individual level of thinking,

provides the opportunity for students to learn by moving beyond what they already know when answering a question, and for the teacher to learn about each student from their attempt,

may have more than one solution or a variety of possible solution paths that range from simple to complex,

requires some degree of cognitive effort (more than just remembering a fact or reproducing a skill),

provides opportunities for students to represent their mathematical ideas using models and written language,

provides opportunities for students to justify their reasoning and evaluate the reasoning of others,

has clear, concise directions.

Can Tasks be Revisited During the Year?

Repeating, or revisiting tasks, allows students to engage with tasks at a deeper level. On the first occasion the student may be focused on 'how to do' the task. Subsequent visits provide an opportunity for students to communicate their mathematical thinking and reasoning more clearly. Making slight variations to a task (e.g. changing the numbers, context, or materials used) will help to maintain interest while students further develop skills and concepts. Some teachers like to introduce tasks whole class and then place tasks in centers for children to revisit at other times throughout the year. Other teachers choose one journal task and repeat it, with slight variations, several times throughout the year as a record of the development of math skills and understandings for student portfolios.

The methods that children use for representing their thinking will change over the course of a year. Repeating a task provides a record of this growth for teachers, parents and students. For example, in Kindergarten an open ended addition task (see work samples below) may be explored early in the year before children begin to write number sentences. Early in the year most kindergarten students will record their thinking in relation to this problem pictorially and may only record one or two solutions to the problem. As the year progresses symbolic representations will gradually begin to appear and representations will become more detailed. The work sample below on the left shows a kindergarten students' attempt to record her thinking early in the school year in response to the task ' Vanessa had 5 cupcakes. Some were chocolate. Some were vanilla. How many were chocolate? How many were vanilla?' Three months later this student completed a similar task, ' Cameron had 6 buttons. Some were green. Some were purple. How many were green? How many were purple?' On this occasion the child's written representation (below right) is more detailed and clearly demonstrates her developing understanding of addition. Although she repeats some number sentences, her drawings show all possible combinations of the six buttons.

How often should I use Math Journals in my class?

Some teachers use several tasks a week as a warm up to the math lesson. Other teachers set aside one period per week for journals, select a task that correlates with the current unit of study and allow more time for students to share their thinking with one another. Tasks may also be used for assessment purposes, or as homework. The important thing is to ensure that students are being given regular opportunities throughout the year to represent their mathematical thinking in ways which makes sense to them.

What type of book should my students use as a Math Journal?

Our experiences in numerous K-5 classrooms have shown that a notebook with blank pages produces the best results. Although these are not always as readily available as ruled notebooks (and are often more expensive) they have a distinct advantage in that students are not restricted by lines and have the space to choose whether to use pictures, numbers, words or a combination of these to record their thinking. Click on the links below to visit our gallery pages to see examples of the types of written responses made by Kindergarten - 5th Grade students when encouraged to make their own decisions about how to record their thinking.

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What are the Characteristics of a Good Math Journal Question? Tasks that ask students to apply memorized procedures to routine problems require one kind of thinking. Non-routine, or open-ended tasks that ask students to engage with concepts and to make connections between ideas and representations require a very different kind of thinking.

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1st Grade - Module 1

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1-10 ten frames

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7-12 dot patterns

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1-6 dot patterns

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The University of Arizona - Institute for Mathematics & Education

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The University of Arizona - Institute for Mathematics & Education

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Cathy Kessel wrote an article with Liping Ma that discuss "unit fraction" vs "fractional unit" and has added a remark to the appendix about this.

Here is the new version.

Hi everybody, Cathy Kessel has compiled all the suggestions that people have made here and other places over the years to produce a final version of the progressions. Here is the compiled document of all the progressions.

And here are two documents outlining the major changes from previous versions.

Note that the University of Arizona has deleted the website that used to host the progressions. I am working to have that url redirect here.

Hi everybody, in order to deal with the spam problem I have deleted all users regarded by WordPress as spam users. These are defined as users who have never posted. Unfortunately this seems to have caught some legitimate users as well. Their content is still on the blog, with the author marked as anonymous.

Cathy Kessel is working on editing the progressions documents into final versions, incorporating comments from this blog and elsewhere and filling in gaps. Here is the fractions progression. We will be rolling out the others over the next few months. Comments welcome!

With the high school writing work in full swing, I've been remiss in posting here lately. I do have some ideas saved up! Meanwhile, Kate Nowak and I have a piece about instructional routines over at the IM blog.

I've been meaning to let you know about the Illustrative Mathematics blog, which launched a few weeks ago. It has blog posts by members of the IM community about our grades 6-8 curriculum and about teaching practice, including a whole series on the 5 practices framework of Smith and Stein. Also, we will be cross posting any IM related posts I write here over there as well. I hope you find our new blog useful!

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SCOE Teacher Support, Modesto, California. 425 likes. Stanislaus County Office of Education's Instructional Support Services provides professional development and assistance for teachers.

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Math Coach / Fact Fluency for Addition and Subtraction

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Math Coach / Fact Fluency for Addition and Subtraction

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Math Coach / Fact Fluency for Addition and Subtraction