Integrating Shapes and Multiplication

We have recently started a Shape and Space unit and I wanted to try and integrate multiplication into questions or tasks that I posed to my students. 

I thought about the idea of making a city using different shapes and different multiplication arrays. 

To start the lesson I showed students the city and then asked them to come up with questions that related to multiplication and shapes.  As always they always come up with much better questions than me.

"How many cubes are in each building?" (that was a very common one)

"What shapes are the buildings?"

"If every cube had two rooms, how many rooms would be in each building?"

"What building is the tallest?"

"If two people lived in every cube, how many people would live in the building?"

Starting a maths lesson by getting students to pose the questions is probably the most enjoyable part of the maths we do in class. It allows me to really see who is thinking mathematically. 

They then had to go off and find out how many cubes are in each building. As this was the start of our Shape and Space unit, we just talked about the shapes of the different buildings. 

The lesson after this. I gave students an opportunity to create their own city. I gave them a success criteria to work from. 

This task really gave the students are great chance to visualise and play with different groupings. 

Later this student used the rulers to level the sides of her building. This brought up the role of right angles.

Students communicating mathematically!

Great that kids are learning how multiplication and addition are often required to help solve problems.

It is always interesting to see what operations the kids come up with. More often than not how I what I would come up with. That confirms the importance of being flexible with numbers or in this case groups of numbers. 

Posing "Challenging" Problems

Posing problems is a appropriate title to this post as they tend to pose me problems trying to write them.

My school was lucky enough to recently have Mignon Weckert talk during a maths job a like. She is passionate about how important posing challenging maths problems are.

So off I went...

The process I went through is based on readings and advice from people who are leaders in inquiry maths and with more knowledge than myself, most notably Mignon.

First, I started with a Central Idea:

Number operations can be used to solve problems in a variety of ways

and with the idea that...
Multiplication and division are effective and efficient ways to solve problems.

From my formative assessment (see previous post) I found that...

• Some students are skip counting
• Some students are using repeated addition
• Some students are using multiplication facts
• Some students are using materials
• Some students are answering it ease!
• Some students have no idea (they told me so!)
• Some students counted all. 

Great 7 different "groups"! 
NOTE: I don't group students any more. The benefits of students working in mixed groups are just too great. I am just aware of where each student is at in their learning.

Now, my first thought was, "How do I pose questions that cater to all of these groups?" 

The advice I was given was..."pose problems that challenge them all!"

Great, but easier said than done!

So, I looked at my "groups" and focused on the central idea. I wanted my student to realise that multiplication and division are efficient and effective methods of solving problems. 

Problem Number 1:

So more by chance than good planning. My class were going on a field trip. I was going to divide them into groups, but I thought... well why don't you let them do it. Now with 21 students the problem immediately presented a challenge as we needed to get into groups of 5. It was also difficult as it dealt with division even before we explore multiplication. I decided to give it a go and see how it went. 

Reflection:

• It was challenging for most, due to the 21 
• To extend those who needed it.  I said each group should have even boys and girls (knowing we have more boys than girls).
• I had rich conversations with my strugglers. We used materials, talked about skip counting or using basic facts.  One student even said that multiplication makes it easier BINGO!
• Some kids drew pictures others used materials.
• I challenged the higher, some even said that it was tricky.
• Some said 4 groups of 5, that didn't work. So we then talked about 5 groups of 4, with 1 more remaining. Commutative properties! 

 Problem Number 2:

I gave students 4 different sets of arrays. Instead of using easy amounts, I made arrays that went over groups of 10, for example 15 groups of 2 or 2 groups of 15. The reason for this was to challenge students to experiment with breaking groups ups (division), combining the use of repeated addition or using skip counting to help solve the problem. The problem was simply to find the totals of each array and show me how.

Reflection:

• Posing harder problems opened up a range of strategies.
• Students worked in groups and learnt from listening to other strategies.
• Students had the flexibility to solve problems in different ways using different operations.
• Students made connections between all 4 operations. 
• It didn't really challenge those higher working students. This is something that I still need to work on. 

My big take away so far is that... that when you make questions challenging, you bring out questions, challenge understandings and encourage discussions because you are making the students think! 

Problem Number 3.

I went to hang out my washing I had 19 pieces of clothing to hang up and 35 pegs. If each piece of clothing needs two pegs do I have enough?

Reflection:

• We discovered the link between addition, subtraction, multiplication and division. i.e. split 19 into 10 and 9 or round up to 20 to help us solve 19 x 2.
• Kids who tried 35 divided by 19 were stumped. Then one said lets use multiplication 19 x 2 as it is easier to solve.
• When working with larger numbers such as 19, it allows you to challenge those who need it and to also support the others. I could work with students on 9 x 2 by creating arrays, but still allowing them to solve the problem. 

Some more questions I have come up...

I have between 15 and 23 apples. How many different ways could I divide them into equal bags?

50 / 5 + 10 = 20 write a story to explain what happens here?

3 X 5 = 30/2 - true or false why?

Use 23 linking cubes to show your understanding the 2, 3, 5 times tables.

Create your own stories using the 2,3,5 times tables.

9 x 0 = 0 Why?

I will continue to add to this post as I move through the unit. If you have any questions that have worked in your class I would love to hear them. 

Provocation Tables for Multiplication and Division

As part of a formative assessment/provocation for the start of our new unit on multiplication and division, I set up four different stations. Each station was designed to provide me with different information to help me plan where to take my students next.

Table 1:

This question was designed to see how students would solve this problem. Students responses varied from...

5+5+5+5+5=24
or
5+5=10, 5+5=10, 10+10=20, 20+5=25
to
5x5=25

Table 2.


This activity was designed to see if students could conceptually visualise multiplication facts.

I really liked this students response. Showing creativity in his maths thinking.


Table 3:

Note: Students knew we have 21 students.

This station was set up to check students understanding of division. I purposely used words such as 'divide' and 'equal groups' to gauge their knowledge of these concepts.

Many students required the use of materials to solve this problem.

Some were about to use their knowledge of multiplication.

Others found it too challenging and wrote 'I don't know?'

Table 4:

For this activity I purposely used the word 'operation'. I wasn't sure if it was too challenging, but I wanted to find out what students knew what it meant. Some required help, but once I had explained it to them, they were then able to provide an answer.

Some students wrote 5x5x5, 2x2x2 and 4x4

Others used addition e.g. 5+5+5

Some students wrote 'I don't know?'

GREAT!!!!!! I always encourage and praise my students for being courageous enough to share that they don't know. I constantly refer to our Maths Essential Agreement that states, "mistakes are good" and "it's ok not to know". Too often maths is seen as a competition to see who can answer questions the fastest, even among 2nd Graders. It is so important that students are able to feel comfortable to share their level of understanding.

An example of a student recording their thinking.

During the activity I asked students to revisit the 'reflection table' to write down any new wonderings they may have.

Pleasing to see students using materials to help them solve problems. It is so important that students don't view using materials as a sign that they don't understand. One student even said, "I am just using them to check"