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Number Line helps students visualize number sequences and demonstrate strategies for counting, comparing, adding, subtracting, multiplying, and dividing.

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Steve Wyborney's Blog: I'm on a Learning Mission.

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Dot Patterns Archives - Steve Wyborney's Blog: I'm on a Learning Mission.

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Color. It's the natural extension to the series that begin with Provide Massive Space to Notice. So, after 180 Opportunities to Notice and How to Create 9 Identical Dot Patterns in 10 Seconds or Less, I'm adding 2 new templates. This time, the circles are blank, so students can use color to record their observations....

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Those you share this with will see the current array and settings.

Make a new rectangle

Change the size of the rectangle by entering whole numbers (up to 50) in the boxes. Then press the MAKE button.

Add and move the sliders

Drag the slider handles from the left or top to divide the rectange into smaller parts. Include up to 4 sliders on each axis.

View partial products

Watch how the equation changes depending on the position of the sliders. Point at any region to see the corresponding piece of the equation.

Show or hide answers

Toggle these buttons to show or hide the results of the equation.

Partial Product Finder allows multiplication combinations to be represented as a rectangle, or array, with dimensions that match the combination. The sides of the rectangle can then be divided into parts to reveal partial products that make up the total product. This helps students develop the flexibility and conceptual understanding required to build fluency with basic facts and strategies for multiplication of larger numbers.

We Welcome Your Feedback

Help us guide the ongoing development of this app by sharing your reactions or suggestions.

Partial Product Finder is brought to you by The Math Learning Center, a nonprofit organization serving the education community since 1976. Visit us at www.mathlearningcenter.org, where you can learn more about our organization, the Bridges in Mathematics PK-5 curriculum we publish, and the many free math education resources we provide, including a robust suite of free virtual manipulatives for use online or as downloadable apps for a variety of operating systems, including Apple iOS and Google Chrome.

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Partial Product, by The Math Learning Center

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Partial Product Finder allows multiplication combinations to be represented as a rectangle, or array, with dimensions that match the combination.

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Share your videos with friends, family, and the world

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I am remaking the ITPs so that they will work on all modern browsers and tablets. They will remain freely available to all without the need for a subscription. This ITP allows you to enter data into a table and then create a line graph to represent the data.

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Visual Times Tables Games

The times tables cards represent multiplication visually, helping children to make connections between multiplication facts. The corresponding PowerPoint games can be used to show children these connections.

Multiplication Dice Pattern Cards

Introductory Games Dice Patterns

Multiplication 10-Frame Cards

Introductory Games 10-Frames

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The times tables cards represent multiplication visually, helping children to make connections between multiplication facts. The corresponding PowerPoint games can be used to show children these connections. Multiplication Dice Pattern Cards Introductory Games Dice Patterns Multiplication 10-Frame Cards Introductory Games 10-Frames

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Introduction

Build up the two times table by combining children's experience of counting in units of two and of representing equal groups; explore how, in a multiplication equation, the factors can appear in either order and the product remains the same.

Teaching points

Teaching point 1: For equally grouped objects, the number of groups is a factor, the group size is a factor, and the overall number of objects is the product; this can be represented with a multiplication equation. Counting in multiples of two can be used to find the product when the group size is two.

Teaching point 2: Counting in multiples of two can be represented by the two times table. Adjacent multiples of two have a difference of two. Facts from the two times table can be used to solve problems about groups of two.

Teaching point 3: Factor pairs can be written in either order, with the product remaining the same (commutativity).

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Times tables: groups of 2 and commutativity (part 1)

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Spine 2: Multiplication and Division - Topic 2.3

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Spine 2: Multiplication and Division – Topic 2.11

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Times tables: 7 and patterns within/across times tables

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Spine 2: Multiplication and Division – Topic 2.9

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Times tables: groups of 10 and of 5, and factors of 0 and 1

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Spine 2: Multiplication and Division – Topic 2.4

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Introduction

Build up the three/six/nine times table; using different structures/interpretations of multiplication and division, solve problems related to these tables; explore connections between the three, six and nine times tables.

Teaching points

Teaching point 1: Counting in multiples of three can be represented by the three times table. Adjacent multiples of three have a difference of three. Facts from the three times table can be used to solve multiplication and division problems with different structures.

Teaching point 2: Counting in multiples of six can be represented by the six times table. Adjacent multiples of six have a difference of six. Facts from the six times table can be used to solve multiplication and division problems with different structures.

Teaching point 3: Products in the six times table are double the products in the three times table; products in the three times table are half of the products in the six times table.

Teaching point 4: Counting in multiples of nine can be represented by the nine times table. Adjacent multiples of nine have a difference of nine. Facts from the nine times table can be used to solve multiplication and division problems with different structures.

Teaching point 5: Products in the nine times table are triple the products in the three times table. Products that are in the three, six and nine times tables share the same factors.

Teaching point 6: Divisibility rules can be used to find out whether a given number is divisible (to give a whole number) by three, six or nine.

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Times tables: 3, 6 and 9, and the relationship between them

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Spine 2: Multiplication and Division - Topic 2.8

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Introduction

Build up the four/eight times table; using different structures/interpretations of multiplication and division, solve problems related to these tables; explore connections between the two, four and eight times tables.

Teaching points

Teaching point 1: Counting in multiples of four can be represented by the four times table. Adjacent multiples of four have a difference of four. Facts from the four times table can be used to solve multiplication and division problems with different structures.

Teaching point 2: Products in the four times table are double the products in the two times table; products in the two times table are half of the products in the four times table.

Teaching point 3: Counting in multiples of eight can be represented by the eight times table. Adjacent multiples of eight have a difference of eight. Facts from the eight times table can be used to solve multiplication and division problems with different structures.

Teaching point 4: Products in the eight times table are double the products in the four times table; products in the four times table are half of the products in the eight times table. Products that are in the two, four and eight times tables share the same factors.

Teaching point 5: Divisibility rules can be used to find out whether a given number is divisible (to give a whole number) by two, four or eight.

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Times tables: 2, 4 and 8, and the relationship between them

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Spine 2: Multiplication and Division - Topic 2.7

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Introduction

Explore how one multiplication equation can have two different grouping interpretations (e.g., an equation from the two times table can be interpreted in terms of groups of two, or two equal groups); make connections between the two times table, doubling and halving.

Teaching points

Teaching point 1: The same multiplication equation can have two different grouping interpretations. Problems about two/five/ten equal groups can be solved using facts from the two/five/ten times table. (commutativity)

Teaching point 2: If two is a factor, knowledge of doubling facts can be used to find the product; problems about doubling can be solved using facts from the two times table.

Teaching point 3: Halving is the inverse of doubling; problems about halving can be solved using facts from the two times table and known doubling facts.

Teaching point 4: Products in the ten times table are double the products in the five times table; products in the five times table are half of the products in the ten times table.

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Commutativity (part 2), doubling and halving

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Spine 2: Multiplication and Division - Topic 2.5

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Number Frames by The Math Learning Center

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Number Frames help students structure numbers to five, ten, twenty, and one hundred. Students use the frames to count, represent, compare, and compute with numbers in a particular range.

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Unleash your creativity with the world's best virtual manipulatives!

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The materials are designed to be used in conjunction with the Numberblocks episodes. They highlight and develop the key mathematical ideas that are embedded in the programmes. Each set of materials comes in the form of a PowerPoint file, and includes the following features:

The Practitioner Notes consist of three elements:

1. The episode description summarises the story and the key things that happen

2. The maths in the episode explains the key mathematical concepts that are featured in the episode

3. Using mathematical language - because it is important that practitioners model precise and correct mathematical language, there are suggestions of key sentences that you might use and have repeated; they provide a language structure to connect each mathematical idea to different contexts. Children will initially use their own language to talk about the mathematics, and will develop correct and precise language if this is modelled by adults.

Talk and Discuss Together provides a set of slides, designed to be used with children after they have watched the episode. They provide a stimulus to talk about what children have noticed, and draw their attention to key aspects of the mathematics. There are often notes for each slide, indicated by a blue dot in the bottom right-hand corner. The notes give ideas for supporting mathematical talk and engaging children in practical activity; some slides have animation to illustrate mathematical concepts.

Enabling Environments gives suggestions for extending the mathematics into the wider Early Years environment where children engage in meaningful activity and explore mathematical concepts.

Learning Together in Year 1 extends the mathematics into Y1. The slides are designed to be used within the context of whole class teaching for mastery.

An overview of each series, their storylines, the mathematics addressed, and links to the episodes on iPlayer, is available as a downloadable PDF. Please be aware that episodes may not always be available on iPlayer:

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Materials to support Early Years and Year 1 teachers

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Too often, we can assume that pupils should be able to get on and engage in independent practice such as revision; however, researchers have shown us how pupils routinely engage in ineffective revision practices. Therefore, pupils benefit from support and explicit teaching of revision strategies, and how to self-regulate their learning.

The Seven-step modelling of revision in practice

The model offers a way to hone evidence-informed revision strategies. Take the commonly used strategies - self-testing flashcards:

1. Activating prior knowledge. Ask pupils when they have used flashcards in the past. When have they found them most useful? Do they remember how much information they should write on them? Will they look different in your subject to other subjects

2. Explicit strategy instruction. Provide step-by-step instructions about how to complete flashcards in your subject - explaining why this is the most effective way to create them and use them (subtle subject specific differences may be easily missed).

3. Modelling of learned strategy. Show an example of a completed flashcard, as well as possibly one that isn't as effective (examples and non-examples) and explain why this is the case. Share your thought processes with the pupils if you complete one live for them, verbalising how you are demonstrating perseverance, resilience and learning from mistakes you have made in the past. For instance, it may be when you have written too much onto a card or dropped a card from the set too early, as you were over-confident.

4. Memorisation of strategy. Check to see if pupils have understood the instructions. You could ask them to explain to their partner why one modelled flashcard is better than another and walk round the room to listen and check for accuracy.

5. Guided practice. Start with some partially completed cards where pupils just need to fill in some missing words, then move on to providing a list of key words for a topic (you could ask the pupils to work in pairs to write the relevant descriptor for them). Some pupils may need the key words for the subsequent topic, whereas others may be able to do this themselves without support.

6. Independent practice. All pupils then complete flashcards for the next topic independently and then try self-testing using the cards.

7. Structured reflection. Individually, or in small groups, support pupils to think about what went well and what they would do differently next time. They could look at each other's cards to think about different techniques (e.g. colour coding) and the possible benefits of this. You may also encourage pupils to reflect on how motivated they were during their self-testing.

Revision will always prove a challenge - perhaps this year as much as any. And yet, explicitly teaching pupils in this way should help enhance the accuracy of their judgements and support them to manage their revision with greater independence.

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it feels like the christmas decorations have only just come down and yet in the blink of an eye the exam season is almost upon us. supporting children to approach exams and to structure their revision effectively is always a challenge

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but this is especially so this year when many pupils are likely to have had periods of absence and disrupted learning.  fortunately

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we have two planning support tools based upon the seven-step model

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from the eef metacognition and self-regulated learning guidance report

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which offer us a useful framework by which to support pupils to structure their exam preparation

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learn how to do so with increasing independence.  the seven-step model the seven-step model - planning worksheet  too often

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we can assume that pupils should be able to get on and engage in independent practice such as revision; however

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researchers have shown us how pupils routinely engage in ineffective revision practices. therefore

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pupils benefit from support and explicit teaching of revision strategies

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and how to self-regulate their learning.    the seven-step modelling of revision in practice  the model offers a way to hone evidence-informed revision strategies. take the commonly used strategies – self-testing flashcards:   activating prior knowledge. ask pupils when they have used flashcards in the past. when have they found them most useful? do they remember how much information they should write on them? will they look different in your subject to other subjects  explicit strategy instruction. provide step-by-step instructions about how to complete flashcards in your subject – explaining why this is the most effective way to create them and use them (subtle subject specific differences may be easily missed).  modelling of learned strategy. show an example of a completed flashcard

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as well as possibly one that isn’t as effective (examples and non-examples) and explain why this is the case. share your thought processes with the pupils if you complete one live for them

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verbalising how you are demonstrating perseverance

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resilience and learning from mistakes you have made in the past. for instance

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it may be when you have written too much onto a card or dropped a card from the set too early

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as you were over-confident.  memorisation of strategy. check to see if pupils have understood the instructions. you could ask them to explain to their partner why one modelled flashcard is better than another and walk round the room to listen and check for accuracy.  guided practice. start with some partially completed cards where pupils just need to fill in some missing words

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then move on to providing a list of key words for a topic (you could ask the pupils to work in pairs to write the relevant descriptor for them). some pupils may need the key words for the subsequent topic

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whereas others may be able to do this themselves without support.  independent practice. all pupils then complete flashcards for the next topic independently and then try self-testing using the cards.  structured reflection. individually

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support pupils to think about what went well and what they would do differently next time. they could look at each other’s cards to think about different techniques (e.g. colour coding) and the possible benefits of this. you may also encourage pupils to reflect on how motivated they were during their self-testing.  revision will always prove a challenge - perhaps this year as much as any. and yet

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explicitly teaching pupils in this way should help enhance the accuracy of their judgements and support them to manage their revision with greater independence.  featured resources   the seven-step model  the seven-step model - planning worksheet

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Supporting Revision and the 'Seven-step Model'

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The EEF's Julie Kettlewell explains how teachers can scaffold pupils' revision using the seven-step model.

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Article by Jenni Way

Good questioning techniques have long being regarded as a fundamental tool of effective teachers. Unfortunately, research shows that 93% of teacher questions are "lower order" knowledge based questions focusing on recall of facts (Daines, 1986). Clearly this is not the right type of questioning to stimulate the mathematical thinking that can arise from engagement in open problems and investigations. Many Primary teachers have already developed considerable skill in good questioning in curriculum areas such as Literacy and History and social studies, but do not transfer these skills to Mathematics. Teachers' instincts often tell them that they should use investigational mathematics more often in their teaching, but are sometimes disappointed with the outcomes when they try it. There are two common reasons for this. One is that the children are inexperienced in this approach and find it difficult to accept responsibility for the decision making required and need a lot of practise to develop organised or systematic approaches. The other reason is that the teachers have yet to develop a questioning style that guides, supports and stimulates the children without removing the responsibility for problem-solving process from the children.

Types of Questions

Within the context of open-ended mathematical tasks, it is useful to group questions into four main categories (Badham, 1994). These questions can be used be the teacher to guide the children through investigations while stimulating their mathematical thinking and gathering information about their knowledge and strategies.

1. Starter questions
These take the form of open-ended questions which focus the children's thinking in a general direction and give them a starting point. Examples:
How could you sort these.......?
How many ways can you find to ....... ?
What happens when we ......... ?
What can be made from....?
How many different ....... can be found?

2. Questions to stimulate mathematical thinking
These questions assist children to focus on particular strategies and help them to see patterns and relationships. This aids the formation of a strong conceptual network. The questions can serve as a prompt when children become 'stuck'. (Teachers are often tempted to turn these questions into instructions, which is far less likely to stimulate thinking and removes responsibility for the investigation from the child).
Examples:
What is the same?
What is different?
Can you group these ....... in some way?
Can you see a pattern?
How can this pattern help you find an answer?
What do think comes next? Why?
Is there a way to record what you've found that might help us see more patterns?
What would happen if....?

3. Assessment questions
Questions such as these ask children to explain what they are doing or how they arrived at a solution. They allow the teacher to see how the children are thinking, what they understand and what level they are operating at. Obviously they are best asked after the children have had time to make progress with the problem, to record some findings and perhaps achieved at least one solution.
Examples:
What have you discovered?
How did you find that out?
Why do you think that?
What made you decide to do it that way?

4. Final discussion questions
These questions draw together the efforts of the class and prompt sharing and comparison of strategies and solutions. This is a vital phase in the mathematical thinking processes. It provides further opportunity for reflection and realisation of mathematical ideas and relationships. It encourages children to evaluate their work.
Examples:
Who has the same answer/ pattern/ grouping as this?
Who has a different solution?
Are everybody's results the same?
Why/why not?
Have we found all the possibilities?
How do we know?
Have you thought of another way this could be done?
Do you think we have found the best solution?

Levels of Mathematical Thinking

Another way to categorise questions is according to the level of thinking they are likely to stimulate, using a hierarchy such as Bloom's taxonomy (Bloom, 1956). Bloom classified thinking into six levels: Memory (the least rigorous), Comprehension, Application, Analysis, Synthesis and Evaluation (requiring the highest level of thinking). Sanders (1966) separated the Comprehension level into two categories, Translation and Interpretation, to create a seven level taxonomy which is quite useful in mathematics. As you will see as you read through the summary below, this hierarchy is compatible with the four categories of questions already discussed.

1. Memory: The student recalls or memorises information
2. Translation: The student changes information into a different symbolic form or language
3. Interpretation: The student discovers relationships among facts, generalisations, definitions, values and skills
4. Application: The student solves a life-like problem that requires identification of the issue and selection and use of appropriate generalisations and skills
5. Analysis: The student solves a problem in the light of conscious knowledge of the parts of the form of thinking.
6. Synthesis: The student solves a problem that requires original, creative thinking
7. Evaluation: The student makes a judgement of good or bad, right or wrong, according to the standards he values.

Combining the Categories

The two ways of categorising types of questions overlap and support each other.

For example, the questions:
Can you see a pattern?
How can this pattern help you find an answer? relate to Interpretation, and;

the questions:
What have you discovered?
How did you find that out?
Why do you think that? require Analysis, and;

the questions:
Have we found all the possibilities?
How do we know?
Have you thought of another way this could be done?
Do you think we have found the best solution? encourage Evaluation.

In the process of working with teachers on this topic, a table was developed which provides examples of generic questions that can be used to guide children through a mathematical investigation, and at the same time prompt higher levels of thinking.

You may also find Jennie Pennant's article Developing a Classroom Culture That Supports a Problem-solving Approach to Mathematics useful, which includes a section on questioning.

References

Badham, V. (1994) What's the Question?. Pamphlet 23. Primary Association for Mathematics (Australia)
Badham, V. (1996). Developing Mathematical Thinking Through Investigations. PAMphlet 31. Primary Association for Mathematics (Australia)
Bloom, B. (1956). Taxonomy of Educational Objectives Handbook 1: Cognitive Domain. New York: David Mackay
Dains, D. (1986). Are Teachers Asking the Right Questions? Education 1, 4 p. 368-374.
Sanders, N. (1966). Classroom Questions: What Kind? New York: Harper and Row.

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Using Questioning to Stimulate Mathematical Thinking

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Good questioning techniques have long being regarded as a fundamental tool of effective teachers. This article for teachers looks at different categories of questions that can promote mathematical thinking.

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TeacherLED - Interactive Whiteboard and Tablet Resources For Teachers

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A selection of easy-to-use classic teaching resources. Includes 100 square, multiplication grid, geobard, digital and analogue clocks, and magnetic numbers

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Our online teacher tools include a wide range of interactive manipulatives and assessment tools to teach and assess basic math concepts. Additional teacher tools help teach literacy and reading skills. Plus, check out our online timers, spinners, and dice that assist with classroom management.

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Teacher Tools " Toy Theater | Learn * Create * Play

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Our online teacher tools include a wide range of interactive manipulatives and assessment tools to teach and assess basic math concepts. Additional teacher tools help teach literacy and reading skills. Plus, check out our online timers, spinners, and dice that assist with classroom management.

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Books are a great way of introducing children to new mathematical concepts. Picture books provide an entertaining and engaging context for practising recall of number facts, such as adding and counting in the Early Years. They can also encourage pupils to solve real-life problems, thinking about how they could help the characters in the story. For older children, books can link together mathematical ideas and provide a rich environment for thinking in more depth about different areas of maths.

Please click on a box in the table below to see some of our recommended books for each age group and mathematical topic. More book suggestions can be found in the list created by the Early Childhood Mathematics Group.

If you're a parent and would like more information about the maths that your child is learning at school, take a look at the book 'Maths for Mums and Dads' by Rob Eastaway and Mike Askew.

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Books are a great way of introducing children to new mathematical concepts. Picture books provide an entertaining and engaging context for practising recall of number facts, such as adding and counting in the Early Years. They can also encourage pupils to solve real-life problems, thinking about how they could help the characters in the story.