This unit follows naturally from the Smart Doubling unit.
In this unit students are encouraged to further develop part/whole mental methods by using the strategy of "make a ten".
- Demonstrate automatic recall of all pairs of single digit addition facts whose total is 10 or less.
- Use the mental strategy "make a ten" for addition problems.
- Use the most efficient mental strategy for a given problem.
In this unit students are encouraged to add to their use of part/whole with doubles by using make a ten methods.
Examples of part/whole methods using make a ten:
- The students work out 8 + 5 by removing 2 from the 5 to leave 3, add this 2 to the 8 to make 10 then add the 3 to the 10 to give 13.
- The students work out 38 + 8 by removing 2 from the 8 to leave 6, add the 2 to the 38 to give 40, then add 6 to the 40 to give 46.
It is desirable for students to move to part/whole methods as counting methods fail for larger numbers. For example, a student who attempts to work out 36 + 46 by counting on will soon lose their way, whereas the part/whole thinker could solve this by adding 30 and 40 to give 70, then add 6 and 6 to get 12 then add 12 to 70 to get 82.
Students who have successfully understood the part/whole methods in the Smart Doubling unit will have little trouble learning the make a ten part/whole strategy.
The learning opportunities in this unit can be differentiated by providing or removing support to students and varying the task requirements. Ways to support students include:
- Introduce fun names for compatible numbers to 10 to encourage engagement. For example, call them 'love’ pairs. 8 loves 2 because they make the special number 10. Numbers can only be with a partner that helps them make 10.
- Vary the way the 10 frames are presented. Children should become used to seeing them vertically or horizontally. When presented horizontally they can represent their left and right fingers.
- A focus on equality so children have a clear understanding that 9 + 8 is equal to 10 + 7
- Relate the strategy ‘make 10’ to higher numbers. For example, 8 + 2 = 10, 18 + 2 = 20, 58 + 2 = 60. Have the children notice what is happening to the pattern. What would the answer be to 138 + 2? What would happen if we had 18 + 12 or 58 + 12?
The context for this unit can be adapted to suit the interests and experiences of your students. For example:
- Recall counting in Maori to 10 (or 20). Ask the class if they can count to 10 (or 20) in another language.
- Make a wall chart with the different ways to say add and subtract in other languages.
- Sets of counters
- Empty tens frames with squares large enough to contain counters
- Tens Frames
- Check the students’ knowledge of the addition facts that add to ten facts before starting. This unit builds on the students’ knowledge of the known facts of ten, i.e., 1 + 9 = 10, 2 + 8 = 10, 3 + 7 = 10, 4 + 6 = 10, 5 + 5 etc.
- Also check that the students understand the " teen" numbers. For example, show the students a ten and a four on tens frame as shown. They should respond that 10 + 4 is 14 without needing to count-on by ones.
- Pose an addition problem where one of the numbers is just under 10:
Anaru has 9 pink lollies and 6 yellow lollies. How many lollies does he have altogether?
Ask the students to model this with counters on tens frames.
- Ask the students to think about ways that they could make this problem quick (or easy) to solve. Encourage the students to think of ways that do not involve counting on by ones.
For example, Anaru moves a yellow lolly to the pink lollies. Now ten and five is shown.
It is important to discuss the fact that the answer to 9 + 6 is the same as the answer to 10 + 5. Without this realisation students will not be able to make progress into mental number processes. Although the problem can of course be correctly solved by counting-on the aim of the lesson is to encourage the development of part/whole mental strategies.
Over the next 3-4 days the students are encouraged to use make to ten strategies for solving number story problems.
- Make up number stories for 9 + 7, 5 + 8, 9 + 8, 7 + 6. Students use counters on empty tens frames and use the technique of filling one of the frames up to show ten counters.
- Students now use the pre-printed tens frames.
Moana has 7 oranges and 9 apples. How many pieces of fruit does she have altogether?
Ask the students to show 7 + 9 with a pre-printed 9 card and a pre-printed 7 card. Students visualise moving 1 out of the 7 to leave 6, and adding 1 to the 9 to create 10. This action of imaging pushes the student towards understanding the movement of numbers around within a problem as a key to mental processing.
- Next, only you, as the teacher, have the pre-printed tens frames. Hold the cards so that only you can see the dots and the students have to visualise what you are seeing. Prior to giving the students addition problems ask the students to first visualise single numbers.
I can see the 8 card. What does it look like?
Some students will "see" the 5 + 3, others will "see" the two spaces.
- Next pose addition problems, which encourage the students to visualise the tens frames that only you can see. For example:
I can see 8+ 6. How could I move the dots to work out 8 + 6?
- The next step in the progression involves no materials. The students visualise the tens frames to move around dots as they solve problems:
8 + 5, 3 + 9, 8 + 9.
- Now link the doubles to the make a ten strategy. Ask the students to consider all the ways they know to work out 8 + 9. The range of likely responses is:
- Double 8 + 1 = 16 + 1 = 17
- Double 9 – 1 = 18 – 1 = 17
- Double 10 – 1 – 2 = 20 – 1 – 2 = 19 – 2 = 17 (Rarely used)
- 10 + 8 = 18 but this is 1 too many so the answer is 17
- Remove 1 from the 8 and add 1 to the 9 and to give 7 + 10 which is 17
- Add 2 to the 8 and remove 2 from the 9 and to give 10 + 7 which is 17
After working through this example together students work in pairs to solve: 6 + 8, 9 + 7, 6 + 9. They discuss their strategies.
- Attention now turns to using make a ten strategies to solve subtraction problems. Subtraction problems with the first number in the teen decade are given.
Michael has 13 lollies and he eats 5 of them. How many lollies does Michael now have now?
Students model 13 on pre-printed ten frames.
Two equally good methods are likely.
EITHER remove the 3 to leave the 10 then remove a further 2 to leave 8.
OR remove 5 from the 10 to leave 5 and add the 3 to give 8.
Students use pre-printed tens frames to solve number stories for subtraction problems. Use: 14 - 6, 17 - 8, 12 - 8, 17 - 5, 13 - 5, 13 - 7, 14 - 3.
Notice the presence of subtractions like 17 - 5. They prevent the mindset that may develop that make a ten is always appropriate. In fact, in this example, breaking the ten is not necessary because 7 - 5 = 2 and add the 10 back in gives 12.
- Now work on using a strategy than involves solving subtraction problems by adding.
What is 13 - 9?
Charlotte thinks of a way to work out 13 - 9 by adding. She asks "9 and what makes 13?" and comes up with the answer 4.
Ask the students how they think Charlotte does this. (She has gone 9 + 1= 10, she remembers the 1, she goes 10 + 3 = 13 and adds the 1 to the 3 to get 4.)
Have the students use Charlotte’s addition method to work out these problems: 12 – 9, 14 – 8, 17 – 14, 16 - 7
- Have students explore both ways they have learnt to solve subtraction problems by using either an addition and a subtraction way to work these problems: 13 - 9, 13 - 4, 17 - 15, 17 - 2, 19 - 17 , 12 – 8.
At the end of each session encourage the students to share and discuss answers as a group or class.