Summer Math Learning

Preschool/K1 Math Skills to Practice

Incoming kindergarten students who will use Bridges in Mathematics should be familiar with the following concepts; these are good skills to practice over the summer.

• Counting forward to 20; counting backwards from 10.  

• Reading and writing numerals to 10

• Beginning understanding of ordinal numbers (1st, 2nd, 3rd, etc.)

• Add/subtract within 10 using fingers, objects and/or pictures

• Comparing objects by size, weight, length (bigger/smaller, heavier/lighter, longer/shorter)

• Recognizing and naming basic 2-D shapes (square, circle, triangle, rectangle, rhombus, trapezoid, hexagon)

• Recognizing, copying and extending simple patterns (AABB, ABAB, etc) 

Exploring Kindergarten Skills

Students ready for a challenge can explore any of the skills we will be learning in K2

• count objects and tell how many there are

• write numerals 0-20

• compare numbers or sets of objects to tell which has more or less

• add & subtract quickly and easily to 5

• add & subtract within 10 using objects, fingers, drawings, numbers, or equations

• solve addition and subtraction story problems

• understand that teen numbers are 10 and some more

• identify and describe shapes

• understand length and weight as something that can be measured

Kindergarten Skills Practice

By the end of Kindergarten, students should be able to successfully do all of the following; these are good skills to practice over the summer

• count objects and tell how many there are

• write numerals 0-20

• compare numbers or sets of objects to tell which has more or less

• add & subtract quickly and easily to 5

• add & subtract within 10 using objects, fingers, drawings, numbers, or equations

• solve addition and subtraction story problems

• understand that teen numbers are 10 and some more

• identify and describe shapes

• understand length and weight as something that can be measured

Exploring Grade 1 Skills

Students ready for a challenge can explore any of the skills we will be learning in Grade 1

• solve addition and subtraction story problems

• count by ones and tens to 120 read and write numbers to 120, and represent a number of objects up to 120 with a written numeral

• understand place value by describing what the digits mean in two-digit numbers; use this knowledge to add and subtract

• add two-digit numbers using at least two different strategies and explain how the strategies work

• subtract two-digit numbers that are multiples of 10 using at least two different strategies and explain how the strategies work

• compare two numbers using the symbols >, =, and <

• add and subtract numbers within 10 efficiently and accurately

• read or construct a graph and answer questions about the data

• measure length using non-standard units such as Popsicle sticks, linking cubes, and so on

• tell and write time to the hour and half-hour on analog and digital clock

Grade 1 Skills Practice

By the end of Grade 1 students should be able to successfully do the following; these are great skills to practice over the summer

• count by ones and tens to 120 read and write numbers to 120, and represent a number of objects up to 120 with a written numeral

• understand place value by describing what the digits mean in two-digit numbers; use this knowledge to add and subtract

• add two-digit numbers using at least two different strategies and explain how the strategies work

• subtract two-digit numbers that are multiples of 10 using at least two different strategies and explain how the strategies work

• compare two numbers using the symbols >, =, and <

• add and subtract numbers within 10 efficiently and accurately

• read or construct a graph and answer questions about the data

• measure length using non-standard units such as Popsicle sticks, linking cubes, and so on

• tell and write time to the hour and half-hour on analog and digital clock

Exploring Grade 2 Skills

Students ready for a challenge can explore any of the skills we will be learning in Grade 2

• solve two-step addition and subtraction story problems to 100

• add and subtract to 20; know addition facts to 20 by memory

• read and write 3-digit numbers using numerals, words, and expanded notation (726 = 700 + 20 + 6)

• understand that the three digits of a 3-digit number represent amounts of hundreds, tens, and ones

• use symbols >, =, < to compare two 3-digit numbers

• add and subtract 2-digit numbers accurately and efficiently, and explain strategies for doing so

• add and subtract 3-digit numbers using models, sketches, and/or numbers, and explain strategies for doing so

• estimate and measure length in centimeters and meters, inches and feet.

• divide circles and rectangles into two, three, and four equal parts and describe the parts

• recognize, draw, and analyze 2- and 3-D shapes

• solve money problems involving dollars, quarters, dimes, nickels, and pennies

Grade 3 Skills Practice

By the end of Grade 3 students should successfully be able to do the following; these are great skills to practice over the summer

• use addition, subtraction, multiplication, and division to solve story problems

• solve multiplication and division facts accurately and efficiently

• multiply 1-digit numbers by multiples of 10, e.g., 3 x 70

• add and subtract 2- and 3-digit numbers

• compare, recognize, and generate equivalent fractions and place them on a number line

• estimate and measure in time, liquid volume, and masses of objects

• round numbers to nearest 10 or 100

• tell time to the minute

• estimate and measure liquid volume and mass in metric units

• solve area and perimeter problems

• identify and construct different kinds of quadrilaterals

• sort and classify shapes

Explore Grade 4 Skills

Students ready for a challenge can explore any of the skills we will be learning in Grade 4

• understand factors and multiples, e.g., 1,2,4,5,10, and 20 are all factors of 20, and 20 is a multiple of each of those numbers

• understand that a prime number has only 2 factors—1 and itself, while a composite number has more than 2 factors

• multiply multi-digit numbers using strategies based on place value and properties of operations

• divide 2-digit numbers by 1-digit numbers using strategies based on place value and the relationship between multiplication and division

• read, write, and compare multi-digit numbers and round multi-digit numbers to any place

• add and subtract multi-digit numbers accurately and efficiently

• recognize and generate equivalent fractions

• compare two fractions with different numerators and denominators

• add and subtract fractions and mixed numbers with like denominators

• multiply a fraction by a whole number

• write fractions with denominators of 10 or 100 in decimal notation

• compare decimal numbers with digits to the hundredths place

• know the relative sizes of measurement units within one system of units, including metric length, metric mass, customary weight, metric volume, and time

• use formulas for area and perimeter of a rectangle to solve problems

• measure and sketch angles with a protractor

• classify 2-D shapes

• identify and draw lines of symmetry

Grade 4 Skills Practice

By the end of Grade 4, students should successfully be able to do the following; these are great skills to practice over the summer

• understand factors and multiples, e.g., 1,2,4,5,10, and 20 are all factors of 20, and 20 is a multiple of each of those numbers

• understand that a prime number has only 2 factors—1 and itself, while a composite number has more than 2 factors

• multiply multi-digit numbers using strategies based on place value and properties of operations

• divide 2-digit numbers by 1-digit numbers using strategies based on place value and the relationship between multiplication and division

• read, write, and compare multi-digit numbers and round multi-digit numbers to any place

• add and subtract multi-digit numbers accurately and efficiently

• recognize and generate equivalent fractions

• compare two fractions with different numerators and denominators

• add and subtract fractions and mixed numbers with like denominators

• multiply a fraction by a whole number

• write fractions with denominators of 10 or 100 in decimal notation

• compare decimal numbers with digits to the hundredths place

• know the relative sizes of measurement units within one system of units, including metric length, metric mass, customary weight, metric volume, and time

• use formulas for area and perimeter of a rectangle to solve problems

• measure and sketch angles with a protractor

• classify 2-D shapes

• identify and draw lines of symmetry

Explore Grade 5 Skills

Students ready for a challenge can explore any of the skills we will be learning in Grade 5

• use models and strategies to divide 2- and 3-digit numbers by 2-digit numbers, with and without remainders. (Students are not expected to use the standard long division algorithm until sixth grade.)

• add and subtract fractions with unlike denominators

• multiply fractions and divide fractions in simple cases

• measure volume

• explain patterns in the number of zeros and the placement of the decimal point in the answer when multiplying or dividing by powers of 10

• read, write, round, compare, add & subtract, multiply & divide decimals

• multiply multi-digit whole numbers

• locate and graph points on a coordinate plane

Required Fluencies in Math

Required fluencies are critical skills students need to be successful at a given grade level.

Fluency means students have

Flexibility - they understand relationships between numbers and concepts

Accuracy - they are able to find the correct solution

Efficiency - they know multiple ways to solve a problem and choose the one that works best; efficiency is NOT just speed

Appropriate (strategy choice) - they know which strategy to use, instead of just choosing the most familiar strategy

K-5 students should have mastered fact fluency-- understanding the relationships between numbers and the operations we use for computation and calculation.

Middle school students will shift to procedural fluency-- using and applying these facts, algorithms and strategies more deeply and in novel situations.

The daily fact practice in the summer packet will expose students to content across multiple standards to help build their fluency; the additional resources being shared here with families provide more games, activities, and books to help build fluency skills in the areas below.

Required Fact Fluencies from Grades 3-5 

Incoming middle school students who still need practice on FACT FLUENCY for the 4 operations are strongly encouraged to practice these skills.

• Single-digit products and quotients (ALL products from memory by end of Grade 3)

• Add/subtract within 1000

• Add/subtract within 1,000,000

• Multi-digit multiplication

Understanding the Standards for Mathematical Practice 

As students begin middle school, critical thinking skills are becoming increasingly important.  In mathematics, we rely on the Standards of Mathematical Practice (SMP’s), which are a part of our Common Core Standards for Mathematics, to help build and support critical thinking.  Instead of mathematics content (like fractions or geometry), these standards focus on the skills and mindsets that students need to be successful mathematicians.

While we expect students to develop and use these practices as they grow in their mathematical learning beginning in Kindergarten, sometimes it is important to practice them explicitly in our math classes.  Middle schoolers need to think more abstractly and to make deeper connections between ideas, so this is the perfect time to work on these practices again. 

One method for doing so is with logic puzzles and brain teasers-- students get to explore some engaging and fun puzzles that do not seem to have anything to do with mathematics, but that actually develop these very skills, which they will rely upon in their mathematics learning.

You can find more information about the Standards for Mathematical Practice on the following pages, as well as at the links below.

VIDEO: Standards for Mathematical Practice -- Student Friendly Language

HANDOUT: Quick Reference Guide Grades 6-8 Mathematical Practice Standards Mass DESE

Grade 5 Skills Practice 

By the end of Grade 5, students be successful doing the following; these are great skills to practice over the summer

• 5.NBT.A Understand the place value system (read, write, compare decimals to the thousandths).

• 5.NBT.B Perform operations with multi-digit whole numbers and with decimals to hundredths.

• 5.NF.A Use equivalent fractions as a strategy to add and subtract fractions.

• 5.NF.B Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

• 5.MD.C Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

Explore Grade 6 Fluencies

Students ready for a challenge can explore the following skills we will be learning in Grade 6

• 6.NS.B.2 Multi-digit division

• 6.NS.B.3 Multi-digit decimal operations 

Grade 6 Skills Practice

By the end of Grade 6, students should be successful doing the following; these are great skills to practice over the summer

• Understand ratio concepts and use ratio reasoning to solve problems.

• Apply and extend previous understandings of multiplication and division to divide fractions by fractions

• Apply and extend previous understandings of numbers to the system of rational numbers.

• Apply and extend previous understandings of arithmetic to algebraic expressions.

• Reason about and solve one-variable equations and inequalities.

• Represent and analyze quantitative relationships between dependent and independent variables.

Explore Grade 7 Fluencies

Students ready for a challenge should continue practicing their fluency from previous years; good options are to use the problems from Open Middle Math, which is linked in the online resources.

Grade 7 Skills Practice

By the end of Grade 7, students should be successful doing the following; these are great skills to practice over the summer

• Analyze proportional relationships and use them to solve real-world and mathematical problems. 

• Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide 

• rational numbers.

• Use properties of operations to generate equivalent expressions.

• Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Explore Grade 8 Fluencies

Students ready for a challenge should continue practicing their fluency from previous years; good options are to use the problems from Open Middle Math, which is linked in the online resources.